Velocity Sentence Examples

velocity
  • The wind velocity did not exceed 20 km.

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  • The concentration is known, and the conductivity can be measured experimentally; thus the average velocity with which the ions move past each other under the existent electromotive force can be estimated.

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  • When he considered all days irrespective of wind velocity, Mazelle found the influence of temperature obliterated.

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  • Simspon concluded that for a given wind velocity dissipation is practically a linear function of ionization.

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  • The other forms of velocity anemometer may be described as belonging to the windmill type.

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  • After Professor Amund Helfand had, in July 1875, discovered the amazingly great velocity, up to 644 ft.

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  • The second method is in principle extremely simple, consisting merely in multiplying the observed velocity of light by the time which it takes light to travel from the sun to the earth.

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  • The velocity of light (q.v.) has been measured with all the precision necessary for the purpose.

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  • Hence the elementary arc divided by the element of time is the rate of change of velocity of the moving-point, or in other words, the velocity in the hodograph is the acceleration in the orbit.

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  • Rising in the high tablelands or on the slopes of the Drakensberg or Lebombo mountains the rivers in their upper courses have a great slope and a high velocity.

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  • In this way the medium velocity of the current may be diminished, and consequently the quantity of water discharged in a given time must, from the effects of friction, be considerably less than that which is computed from theory.

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  • From a collection of the best experiments by previous workers he selected eighty-two (fifty-one on the velocity of water in conduit pipes, and thirty-one on its velocity in open canals); and, discussing these on physical and mechanical principles, he succeeded in drawing up general formulae, which afforded a simple expression for the velocity of running water.

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  • In particular, for a jet issuing into the atmosphere, where p=P, q 2 /2g = h - z, (9) or the velocity of the jet is due to the head k-z of the still free surface above the orifice; this is Torricelli's theorem (1643), the foundation of the science of hydrodynamics.

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  • At the main base in Adelie Land autumn sledging proved impossible, and throughout the winter there was a continuous succession of terrific blizzards, wind with an average velocity of 50 m.p.h.

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  • The velocity is now well determined; the difficulty is to determine the time of passage.

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  • He supposed that the filaments of water which graze along the sides of the pipe lose a portion of their velocity; that the contiguous filaments, having on this account a greater velocity, rub upon the former, and suffer a diminution of their celerity; and that the other filaments are affected with similar retardations proportional to their distance from the axis of the pipe.

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  • In the Eulerian method the attention is fixed on a particular point of space, and the change is observed there of pressure, density and velocity, which takes place during the motion; but in the Lagrangian method we follow up a particle of fluid and observe how it changes.

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  • I n a straight uniform current of fluid of density p, flowing with velocity q, the flow in units of mass per second across a plane area A, placed in the current with the normal of the plane making an angle 0 with the velocity, is oAq cos 0, the product of the density p, the area A, and q cos 0 the component velocity normal to the plane.

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  • Generally if S denotes any closed surface, fixed in the fluid, M the mass of the fluid inside it at any time t, and 0 the angle which the outward-drawn normal makes with the velocity q at that point, dM/dt = rate of increase of fluid inside the surface, (I) =flux across the surface into the interior _ - f f pq cos OdS, the integral equation of continuity.

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  • A small sphere of the fluid, if frozen suddenly, would retain this angular velocity.

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  • Calling the sum of the pressure and potential head the statical head, surfaces of constant statical and dynamical head intersect in lines on H, and the three surfaces touch where the velocity is stationary.

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  • Thus if d,/ is the increase of 4, due to a displacement from P to P', and k is the component of velocity normal to PP', the flow across PP' is d4 = k.PP'; and taking PP' parallel to Ox, d,, = vdx; and similarly d/ ' = -udy with PP' parallel to Oy; and generally d4,/ds is the velocity across ds, in a direction turned through a right angle forward, against the clock.

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  • The curves 0 = constant and 4, = constant form an orthogonal system; and the interchange of 0 and 4, will give a new state of uniplanar motion, in which the velocity at every point is turned through a right angle without alteration of magnitude.

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  • For instance, in a uniplanar flow, radially inward towards 0, the flow across any circle of radius r being the same and denoted by 27rm, the velocity must be mfr, and 0=m log r,, y=m0, +4,i =m log re ie, w=m log z.

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  • A single vortex will remain at rest, and cause a velocity at any point inversely as the distance from the axis and perpendicular to its direction; analogous to the magnetic field of a straight electric current.

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  • If other vortices are present, any one may be supposed to move with the velocity due to the others, the resultant stream function being = gy m log r =log IIrm; (9) the path of a vortex is obtained by equating the value of 1P at the vortex to a constant, omitting the rm of the vortex itself.

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  • Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it.-A stream-function 4, must be determined to satisfy the conditions v24 =o, throughout the liquid; (I) I =constant, over any fixed boundary; (2) d,t/ds = normal velocity reversed over a solid boundary, (3) so that, if the solid is moving with velocity U in the direction Ox, d4y1ds=-Udy/ds, or 0 +Uy =constant over the moving cylinder; and 4,+Uy=41' is the stream function of the relative motion of the liquid past the cylinder, and similarly 4,-Vx for the component velocity V along Oy; and generally 1,1'= +Uy -Vx (4) is the relative stream-function, constant over a solid boundary moving with components U and V of velocity.

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  • If the liquid is stirred up by the rotation R of a cylindrical body, d4lds = normal velocity reversed dy = - Rx- Ry ds (5) ds 4' + 2 R (x2 + y2) = Y, (6) a constant over the boundary; and 4,' is the current-function of the relative motion past the cylinder, but now V 2 4,'+2R =o, (7) throughout the liquid.

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  • Over a concentric cylinder, external or internal, of radius r=b, 4,'=4,+ Uly =[U I - + Ui]y, (4) and 4" is zero if U 1 /U = (a 2 - b2)/b 2; (5) so that the cylinder may swim for an instant in the liquid without distortion, with this velocity Ui; and w in (I) will give the liquid motion in the interspace between the fixed cylinder r =a and the concentric cylinder r=b, moving with velocity U1.

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  • If the liquid is reduced to rest at infinity by the superposition of an opposite stream given by w = - Uz, we are left with w = Ua2/z, (6) =U(a 2 /r) cos 0= Ua2x/(x2+y2), (7) 4, = -U(a 2 /r) sin 0= -Ua2y/( x2+y2), (8) giving the motion due to the passage of the cylinder r=a with velocity U through the origin 0 in the direction Ox.

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  • If the direction of motion makes an angle 0' with Ox, tan B' = d0 !dam _ ?xy 2 = tan 20, 0 =-10', (9) dy/ y and the velocity is Ua2/r2.

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  • When the cylinder r =a is moved with velocity U and r =b with velocity U 1 along Ox, = U b e - a,1 r +0 cos 0 - U ib2 - 2 a, (r +Q 2 ') cos 0, = - U be a2 a2 (b 2 - r) sin 0 - Uib2 b1)a, (r - ¢2 sin 0; b and similarly, with velocity components V and V 1 along Oy a 2 b2 ?= Vb,_a,(r+r) sin g -Vi b, b2 a, (r+ 2) sin 0, (17) = V b, a2 a, (b2 r) cos 0+Vi b, b, a, (r- ¢ 2) cos h; (18) and then for the resultant motion z 2zz w= (U 2 + V2)b2a a2U+Vi +b a b a2 U z Vi -(U12+V12) b2 z a2b2 Ui +VIi b 2 - a 2 U1 +Vii b 2 - a 2 z The resultant impulse of the liquid on the cylinder is given by the component, over r=a (§ 36), X =f p4 cos 0.ad0 =7rpa 2 (U b z 2 + a 2 Uib.2bz a2); (20) and over r =b Xi= fp?

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  • With v=o, the angular velocity of the cylinder is 2w; in this way the velocity may be calculated of the propagation of ripples and waves on the surface of a vertical whirlpool in a sink.

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  • Another explanation may be given of the sidelong force, arising from the velocity of liquid past a cylinder, which is encircled by a vortex.

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  • The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a resultant thrust on the whole cylinder is 2mU sin Olga, and its thrust is 27rpmU absolute units in the direction Cy, to be counteracted by a support at the centre C; the liquid is streaming past r=a with velocity U reversed, and the cylinder is surrounded by a vortex.

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  • The velocity of a liquid particle is thus (a 2 - b 2)/(a 2 +b 2) of what it would be if the liquid was frozen and rotating bodily with the ellipse; and so the effective angular inertia of the liquid is (a 2 -b 2) 2 /(a 2 +b 2) 2 of the solid; and the effective radius of gyration, solid and liquid, is given by k 2 = 4 (a 2 2), and 4 (a 2 For the liquid in the interspace between a and n, m ch 2(0-a) sin 2E 4) 1 4Rc 2 sh 2n sin 2E (a2_ b2)I(a2+ b2) = I/th 2 (na)th 2n; (8) and the effective k 2 of the liquid is reduced to 4c 2 /th 2 (n-a)sh 2n, (9) which becomes 4c 2 /sh 2n = s (a 2 - b 2)/ab, when a =00, and the liquid surrounds the ellipse n to infinity.

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  • An angular velocity R, which gives components - Ry, Ix of velocity to a body, can be resolved into two shearing velocities, -R parallel to Ox, and R parallel to Oy; and then ik is resolved into 4'1+1'2, such that 4/ 1 -R-Rx 2 and 1//2+IRy2 is constant over the boundary.

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  • In a similar way the more general state of motion may be analysed, given by w =r ch2('-y), y =a+, i, (26) as giving a homogeneous strain velocity to the confocal system; to which may be added a circulation, represented by an additional term in w.

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  • Motion symmetrical about an Axis.-When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function ' can be found analogous to that employed in plane motion, such that the flux across the surface generated by the revolution of any curve AP from A to P is the same, and represented by 2s-4 -11'o); and, as before, if d is the increase in due to a displacement of P to P', then k the component of velocity normal to the surface swept out by PP' is such that 274=2.7ryk.PP'; and taking PP' parallel to Oy and Ox, u= -d/ydy, v=dl,t'/ydx, (I) and 1P is called after the inventor, " Stokes's stream or current function," as it is constant along a stream line (Trans.

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  • The vortex advances with a certain velocity; and if an equal circular vortex is generated coaxially with the first, the mutual influence can be observed.

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  • The components of velocity of the moving origin are denoted by U, V, W, and the components of angular velocity of the frame of reference by P, Q, R; and then if u, v, w denote the components of fluid velocity in space, and u', v', w' the components relative to the axes at a point (x, y, z) fixed to the frame of reference, we have u =U +u' - yR +zQ, v =V +v -zP +xR, w=W +w -xQ +yP.

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  • Thus, for example, with = 4Uy 2 (r 2 a 2 -I), r2 = x2 +y 2, (13) for the space inside the sphere r=a, compared with the value of, i' in § 34 (13) for the space outside, there is no discontinuity of the velocity in crossing the surface.

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  • Hill's spherical vortex, advancing through the surrounding liquid with uniform velocity.

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  • As an application of moving axes, consider the motion of liquid filling the ellipsoidal case 2 y 2 z2 Ti + b1 +- 2 = I; (1) and first suppose the liquid be frozen, and the ellipsoid l3 (4) (I) (6) (9) (I o) (II) (12) (14) = 2 U ¢ 2, (15) rotating about the centre with components of angular velocity, 7 7, f'; then u= - y i +z'i, v = w = -x7 7 +y (2) Now suppose the liquid to be melted, and additional components of angular velocity S21, 522, S23 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function 2224_ - S2 b c 6 a 5 x b2xy, as may be verified by considering one term at a time.

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  • To determine the motion of a jet which issues from a vessel with plane walls, the vector I must be constructed so as to have a constant (to) (II) the liquid (15) 2, integrals;, (29) (30) (I) direction 0 along a plane boundary, and to give a constant skin velocity over the surface of a jet, where the pressure is constant.

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  • The stream lines xBAJ, xA'J' are given by = 0, m; so that if c denotes the ultimate breadth JJ' of the jet, where the velocity may be supposed uniform and equal to the skin velocity Q, m=Qc, c=m/Q.

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  • Ja - u  ?I a -a b -u' sh nS2=sh log (Q)=?a - b a - a' b - u' At x where = co, u = o, and q= go, (O n b - a ' a + a -b a' cio) - ?a-a'?b a-a' q In crossing to the line of flow x'A'P'J', b changes from o to m, so that with q = Q across JJ', while across xx the velocity is qo, so that i n = go.

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  • The motion of a jet impinging on an infinite barrier is obtained by putting j = a, j' = a'; duplicated on the other side of the barrier, the motion reversed will represent the direct collision of two jets of unequal breadth and equal velocity.

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  • The continuity is secured if the liquid between two ellipsoids X and X 11 moving with the velocity U and 15 1 of equation (II), is squeezed out or sucked in across the plane x=o at a rate equal to the integral flow of the velocity I across the annular area a l.

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  • When the liquid is bounded externally by the fixed ellipsoid A = A I, a slight extension will give the velocity function 4 of the liquid in the interspace as the ellipsoid A=o is passing with velocity U through the confocal position; 4 must now take the formx(1'+N), and will satisfy the conditions in the shape CM abcdX ¢ = Ux - Ux a b x 2+X)P Bo+CoB I - C 1 (A 1 abcdX, I a1b1cl - J o (a2+ A)P and any'confocal ellipsoid defined by A, internal or external to A=A 1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox BA+CA-B 1 -C1 W'.

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  • A distribution of sources and doublets over a moving surface will enable an expression to be obtained for the velocity function of a body moving in the presence of a fixed sphere, or inside it.

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  • The partial differential coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or angular, which corresponds.

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  • Conversely, if the kinetic energy T is expressed as a quadratic function of x, x x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.

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  • These theorems, which hold for the motion of a single rigid body, are true generally for a flexible system, such as considered here for a liquid, with one or more rigid bodies swimming in it; and they express the statement that the work done by an impulse is the product of the impulse and the arithmetic mean of the initial and final velocity; so that the kinetic energy is the work done by the impulse in starting the motion from rest.

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  • It is driven by a powerful engine through triple gearing of 42 to 1, and speeded to have a surface velocity of rollers of 15 ft.

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  • By the mode of admission the hot liquor at its entry is distributed over a large area relatively to its volume, and while this is necessarily effected with but little disturbance to the contents of the vessel, a very slow velocity is ensured for the current of ascending juice.

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  • The slower class of meteors overtaking the earth (like the Andromedids of November) have a velocity of about 8 or 10 m.

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  • The meteors move very slowly, as they have to overtake the earth, and their apparent velocity is only about 9 m.

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  • Similar condensations produced the sun and stars; and the flaming state of these bodies is due to the velocity of their motions.

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  • A comparison of the force habitually developed by the wind in various parts of the islands shows that at Suttsu in Yezo the average strength is 9 metres per second, while Izuhara in the island Tsu-shima, Kumamoto in KiOshi and Gifu in the east centre of the main island stand at the bottom of the list with an average wind velocity of only 2 metres.

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  • Their season is from June to October, but they occur in other months also, and they develop a velocity of 5 to 75 m.

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  • An exactly similar expression holds good in hydrokinetics, provided that for the electric potential we substitute velocity potential, and for the electric force the velocity of the liquid.

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  • These are paved with stone blocks or lined with mercury riffles, so that from the greatly reduced velocity of flow, due to the sudden increase of surface, the finer particles of gold may collect.

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  • In 1628 Castelli published a small work, Della misura dell' acque correnti, in which he satisfactorily explained several phenomena in the motion of fluids in rivers and canals; but he committed a great paralogism in supposing the velocity of the water proportional to the depth of the orifice below the surface of the vessel.

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  • This increase of velocity implies an increase of the reaction on the surface, the black side of a vane being thus pressed with greater force than the bright side.

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  • In later memoirs Reynolds followed up this subject by proceeding to establish definitions of the velocity and the momentum and the energy at an element of volume of the molecular medium, with the precision necessary in order that the dynamical equations of the medium in bulk, based in the usual manner on these quantities alone, without directly considering thermal stresses, shall be strictly valid - a discussion in which the relation of ordinary molar mechanics to the more complete molecular theory is involved.

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  • He had learnt from Torstensson that Denmark was most vulnerable if attacked from the south, and, imitating the strategy of his master, he fell upon her with a velocity which paralysed resistance.

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  • To remedy drawback (2) Repsolds provided for the Yale heliometer an additional handle for motion in position angle, intermediate in velocity between the original quick and slow motions.

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  • The difference between the position as determined astronomically and by dead-reckoning gives an excellent idea of the general direction and velocity of the surface currents.

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  • This deflecting force is directly proportional to the velocity and the mass of the particle and also to the sine of the latitude; hence it is zero at the equator and comes to a maximum at the poles.

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  • The deeper layers lag behind the upper in deflection and the velocity of the current rapidly diminishes in consequence.

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  • He called the depth at which the opposed direction is attained the driftcurrent depth, and he found it to be dependent on the velocity of the surface current and on the latitude.

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  • By the use of the spiral guide casing and the chimney the velocity of the effluent air is gradually FIG.

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  • The angular velocity of the shaft is proportional to the rate of working.

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  • A number of molecules moving in obedience to dynamical laws will pass through a series of configurations which can be theoretically determined as soon as the structure of each molecule and the initial position and velocity of every part of it are known.

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  • Each impinging molecule exerts an impulsive pressure equal to mu on the boundary before the component of velocity of its centre of gravity normal to the boundary is reduced to zero.

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  • A particle of this mass is easily visible microscopically, and a velocity of 2 mm.

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  • But it can be shown that from the aggregation of these separate short motions the particle ought to have a resultant motion, described with an average velocity which, although much smaller than 2 mm.

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  • Modern research has proved that such reactions are not occasioned by water acting as H 2 0, but really by its ions (hydrions and hydroxidions), for the velocity is proportional (in accordance with the law of chemical mass action) to the concentration of these ions.

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  • It passes over equal spaces in equal times, but falls with an accelerating velocity according to the formula h = zgt 2, where h is the height fallen through, g the force of gravity, and t the time of flight.

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  • The yard scales were on detachable strips, so that fresh strips could be inserted for variations in velocity.

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  • C. Vogel's spectroscopic measures in 1889.2 Previously to each obscuration, the star was found to be moving rapidly away from the earth; its velocity then diminished to zero pari passu with the loss of light, and reversed its direction during the process of recovery.

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  • If each wave travels out from the source with velocity U the n waves emitted in one second must occupy a length U and therefore U = nX.

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  • The distribution of velocity then is represented by the dotted curve and is forward when the curve is above the axis and Dackward when it is below.

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  • To find the relation of the velocity to displacement and pressure we shall express the fact that the wave travels on carrying all its conditions with it, so that the displacement now at M will arrive at N while the wave travels over MN.

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  • Then u/U = - dy/dx (2) This gives the velocity of any particle in terms of the displacement.

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  • Equating (I) and (2) u/U = wÆ (3) which gives the particle velocity in terms of the pressure excess.

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  • Generally, if any condition in the wave is carried forward unchanged with velocity U, the change of 4 at a given point in time dt is equal to the change of as we go back along the curve a distance dx = Udt at the beginning of dt.

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  • It is convenient to give this calculation before proceeding to describe the experimental determination of the velocity in air, in other gases and in water, since the calculation serves to some extent as a guide in conducting and interpreting the observations.

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  • Every particle in the plane will have the same displacement and the same velocity, and these will be perpendicular to the plane and parallel to the line of propagation.

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  • Whatever the form of a wave, we could always force it to travel on with that form unchanged, and with any velocity we chose, if we could apply any " external " force we liked to each particle, in addition to the " internal " force called into play by the compressions or extensions.

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  • But it has velocity U, and therefore momentum poU 2 is carried in.

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  • If the velocity of a particle at A relative to the undisturbed parts is u from left to right, the velocity of the matter moving out at A is U - u, and the momentum carried out by the moving matter is p(U - u) 2.

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  • If then we apply a pressure X given by (5) at every point, and move the medium with any uniform velocity U, the disturbance remains fixed in space.

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  • Or if we now keep the undisturbed parts of the medium fixed, the disturbance travels on with velocity U if we apply the pressure X at every point of the disturbance.

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  • If the velocity U is so chosen that E - poU 2 = o, then X = o, or the wave travels on through the action of the internal forces only, unchanged in form and with velocity U = (E/p).

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  • If we omitted it we should have to assume this, and equation (6) would give us the velocity of propagation if the assumption were justified.

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  • If, however, we put on external forces of the required type X it is obvious that any wave can be propagated with any velocity, and our investigation shows that when U has the value in (6) then and only then X is zero everywhere, and the wave will be propagated with that velocity when once set going.

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  • He supposed that in air Boyle's law holds in the extensions and compressions, or that p = kp, whence dp/dp = k = p/p. His value of the velocity in air is therefore U = iJ (p ip.) (Newton's formula).

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  • But for very small times the assumption may perhaps be made, and the result at least shows the way in which the velocity is affected by the addition of a small term depending on and changing sign with u.

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  • We see at once that, where u=o, the velocity has its " normal " value, while where u is positive the velocity is in excess, and where u is negative the velocity is in defect of the normal value.

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  • In ordinary sound-waves the effect of the particle velocity in affecting the velocity of transmission must be very small.

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  • The maximum particle velocity is 21rna (where n is the frequency and a the amplitude), or 27raU/X.

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  • But there is no doubt that with very loud explosive sounds the normal velocity is quite considerably exceeded.

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  • This is hardly to be explained by equation (I I), for at the very front of the disturbance u =o and the velocity should be normal.

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  • The kinetic energy per cubic centimetre is 2 pu t, where is the density and u is the velocity of disturbance due to the passage of the wave.

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  • An obvi us method of determining the velocity of sound in air consists in starting some sound, say by firing a gun, and stationing an observer at some measured distance from the gun.

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  • The distance divided by the time gives the velocity of the sound.

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  • Regnault in the years 1862 to 1866 on the velocity of sound in open air, in air in pipes and in various other gases in pipes, he sought to eliminate personal equaticn by dispensing with the human element in the observations, using electric receivers as observers.

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  • On page 459 of the Memoire will be found a list of previous careful experiments on the velocity of sound.

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  • The temperature of the air traversed and its humidity were observed, and the result was finally corrected to the velocity in dry air at o C. by means of equation (ro).

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  • In the memoir cited above Regnault gives an account of determinations of the velocity in air in pipes of great length and of diameters ranging from o 108 metres to i i metres.

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  • He found that in all cases the velocity decreased with a diameter.

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  • The sound travelled to and fro in the pipes several times before the signals died away, and he found that the velocity decreased with the intensity, tending to a limit for very feeble sounds, the limit being the same whatever the source.

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  • He found that within wide limits the velocity was independent of the pressure, thus confirming the theory.

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  • Correcting the velocity obtained in the 0 .

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  • They found that the velocity of propagation of different musical sounds was the same.

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  • When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.

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  • When a wave of sound travelling through one medium meets a second medium of a different kind, the vibrations of its own particles are communicated to the particles of the new medium, so that a wave is excited in the latter, and is propagated through it with a velocity dependent on the density and elasticity of the second medium, and therefore differing in general from the previous velocity.

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  • As with light the ratio involved in the second law is always equal to the ratio of the velocity of the wave in the first medium to the velocity in the second; in other words, the sines of the angles in question are directly proportional to the velocities.

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  • Hence sound rays, in passing from one medium into another, are bent in towards the normal, or the reverse, according as the velocity of propagation in the former exceeds or falls short of that in the latter.

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  • It further follows, as in the analogous case of light, that there is a certain angle termed the critical angle, whose sine is found by dividing the less by the greater velocity, such that all rays of sound meeting the surface separating two different bodies will not pass onward, but suffer total reflection back into the first body, if the.

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  • On the other hand, to produce convergence with water or hydrogen gas, in both which the velocity of sound exceeds its rate in air, the lens ought to be concave.

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  • Now if the temperature is higher overhead than at the surface, the velocity overhead is greater.

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  • But usually the lower layers are warmer than the upper layers, and the velocity below is greater than the velocity above.

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  • Stokes showed that this effect is one of refraction, due to variation of velocity of the air from the surface upwards Brit.

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  • It is, of course, a matter of common observation that the wind increases in velocity from the surface upwards.

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  • The velocity of any part of a wave front relative to the ground will be the normal velocity of sound + the velocity of the wind at that point.

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  • Since the velocity increases as we go upwards the front tends to swing round and travel downwards, as shown in the successive positions a I, 2, 3 and 4, in fig.

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  • But if the wind is against the sound the velocity of a point of the wave front is the normal velocity-the wind velocity at the point, and so decreases as we rise.

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  • This will go on continually as long as air is supplied to the cylinder, and the velocity of rotation of the upper plate will be accelerated up to a certain maximum, at which it may be maintained by keeping the force of the current constant.

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  • The result is a note whose pitch rises as the velocity of rotation increases, and becomes steady when that velocity reaches its constant value.

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  • Since U=n X where U is the velocity of sound, X the wave-length, and n the frequency, it follows that the forward frequency is greater than the backward frequency.

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  • Let S' be its position one second later, its velocity being u.

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  • Let R be the receiver at a given instant, R' its position a second later, its velocity being v.

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  • Let the velocity of the air from S to R be w, and let U be the velocity of sound in still air.

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  • If now the curve moves along unchanged in form in the direction ABC with uniform velocity U, the epoch e =OA at any time t will be Ut, so that the value of y may be represented as 2 y=a sin T (x - Ut).

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  • Then, as we shall prove later, the vibrations of the string may be represented by the travelling of two trains in opposite directions each with velocity /tension=mass per unit length each half the height of the train represented in fig.

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  • The maximum velocity of a particle in the wave-train is the amplitude of dy/dt.

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  • Since the velocity is the same for all disturbances they all travel at the same speed, and the two trains will always remain of the same form.

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  • We see, then, that the conditions for the application of Fourier's theorem are equivalent to saying that all disturbances will travel along the system with the same velocity.

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  • If U 4 is the velocity of longitudinal waves along the sounder, and 1 the length of the sounder, the frequency of vibration is U 8 /2l.

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  • If L 1 is the internodal distance and U 1 the velocity in a gas, L and U being the corresponding values for air, we have U 1 /U =L1/L.

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  • If U is the velocity of sound in a gas at pressure P with density p, and if waves of length X and frequency N are propagated through it, then the distanc?e l between the dust-heaps is 2 = N - zN Vyp' where y is the ratio of the two specific heats.

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  • We shall first investigate the velocity with which a disturbance travels along a string of mass m per unit length when it is stretched with a constant tension T, the same at all points.

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  • Then move AB from right to left with this velocity, and the disturbance remains fixed in space.

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  • We shall find the velocity of propagation, just as in previous cases, from the consideration of transfer of momentum.

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  • Suppose that a disturbance is travelling with velocity U unchanged in form along a rod from left to right.

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  • At B there is only the latter kind, and since the transfer of matter is powoU, where po is the undisturbed density and wo is the undisturbed cross-section, since its velocity is U the passage of momentum per second is powoUo 2.

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  • At A, if the velocity of the disturbance relative to undisturbed parts of the rod is u from left to right, the velocity relative to A is U - u.

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  • The velocity with which the rod must travel in order that the disturbance may be fixed in space is therefore U =, I (Y/p), or, if the rod is kept fixed, this is the velocity with which the disturbance travels.

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  • But keeping r/X small we may as before form stationary waves, and it is evident that the series of fundamental and overtones will be just as with the air in pipes, and we shall have the same three types - fixed at one end, free at both ends, fixed at both ends - with fundamental frequencies respectively 41, p ' 21 V p, and I velocity in rod =velocity in air X distance between dust heaps.

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  • The velocity of propagation of a torsional disturbance along a wire of circular section may be found by the transfer of momentum method, remembering that we must now replace linear momentum by angular momentum.

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  • If 0 is the angle of twist, the angular velocity is d0/dt.

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  • The velocity of a disturbance along such a bar, and its modes of vibration, depend therefore on the elastic properties of the material and the dimensions of the bar.

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  • Substituting in (33) we get U 2 = n/p. (34) If we now keep the wire at rest the disturbance travels along it with velocity U= d (nip), and it depends on the rigidity and density of the wire and not upon its radius.

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  • When the velocity of the jet is gradually increased there is a certain range of velocity for which the jet is unstable, so that any deviation from the straight rush-out tends to increase as the jet moves up. If then the jet is just on the point of instability, and is subjected as its base to alternations of motion, the sinuosities impressed on the jet become larger and larger as it flows out, and the flame is as it were folded on itself.

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  • But, if quite regular disturbances are impressed on the jet at intervals of time which depend on the diameter and speed of outflow (they must be somewhat more than ?r times its diameter apart), these disturbances go on growing and break the stream up into equal drops, which all move with the same velocity one after the other.

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  • The third mode of production of combination tones, the production in the medium itself, follows from the varying velocity of different parts of the wave, as investigated at the beginning of this article.

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  • It is easily shown that after a time we shall have to superpose on the original displacement a displacement proportional to the square of the particle velocity, and this will introduce just the same set of combination tones.

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  • If w is the weight of a locomotive in tons, r the radius of curvature of the track, v the velocity in feet per sec.; then the horizontal force exerted on the bridge is wv 2 /gr tons.

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  • Gounelle measured the velocity of electricity.

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  • Let E be the effective elasticity of the aether; then E = pc t, where p is its density, and c the velocity of light which is 3 X 10 10 cm./sec. If = A cos" (t - x/c) is the linear vibration, the stress is E dE/dx; and the total energy, which is twice the kinetic energy Zp(d/dt) 2 dx, is 2pn2A2 per cm., which is thus equal to 1.8 ergs as above.

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  • If we rest on the synthesis here described, the energy of the matter, even the thermal part, appears largely as potential energy of strain in the aether which interacts with the kinetic energy associated with disturbances involving finite velocity of matter.

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  • But it is found not to vary at all, even up to the second order of the ratio of the earth's velocity to that of light.

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  • We shall make the natural supposition that motion of the aether, say with velocity (u,v,w) at the point (x,y,z), is simply superposed on the velocity V of the optical undulations through that medium, the latter not being intrinsically altered.

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  • If this relation is true along all paths, the velocity of the aether must be of irrotational type, like that of frictionless fluid.

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  • As, however, our terrestrial optical apparatus is now all in motion along with the matter, we must dealt .with the rays relative to the moving system, and to these also Fermat's principle clearly applies; thus V+ (lu'--mv'-Fnw') is here the velocity of radiation in the direction of the ray, but relative to the moving material system.

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  • This theory secures that the times of passage of the rays shall be independent of the motion of the system, only up to the first order of the ratio of its velocity to that of radiation.

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  • Now the electric force (P,Q,R) is the force acting on the electrons of the medium moving with velocity v; consequently by Faraday's electrodynamic law (P,Q,R) = (P',Q' - vc, R'+vb) where (P',Q',R') is the force that would act on electrons at rest, and (a,b,c) is the magnetic induction.

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  • If v varies with respect to locality, or if there is a velocity of convection (p,q,r) variable with respect to direction and position, and analytical expression of the relation (ii) assumes a more complex form; we thus derive the most general equations of electrodynamic propagation for matter treated as continuous, anyhow distributed and moving in any manner.

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  • Trains of waves nearly but not quite homogeneous as regards wave-length will as usual be propagated as wave-groups travelling with the slightly different velocity d(VX-1)/dX-', the value of K occurring in V being a function of X determined by the law of optical dispersion of the medium.

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  • According to these experiments, the resistance of the air can be represented by no simple algebraical law over a large range of velocity.

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  • Abandoning therefore all a priori theoretical assumption, Bashforth set to work to measure experimentally the velocity of shot and the resistance of the air by means of equidistant electric screens furnished with vertical threads or wire, and by a chronograph which measured the instants of time at which the screens were cut by a shot flying nearly horizontally.

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  • As a first result of experiment it was found that the resistance of similar shot was proportional, at the same velocity, to the surface or cross section, or square of the diameter.

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  • The resistance R can thus be divided into two factors, one of which is d 2, where d denotes the diameter of the shot in inches, and the other factor is denoted by p, where p is the resistance in pounds at the same velocity to a similar I-in.

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  • We first determine the time t in seconds required for the velocity of a shot, d inches in diameter and weighing w lb, to fall from any initial velocity V(f/s) to any final velocity v(f/s).

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  • We put and call C the ballistic coefficient (driving power) of the shot, so that (6) At = COT, where (7) AT = Av/gp, and AT is the time in seconds for the velocity to drop Av of the standard shot for which C = I, and for which the ballistic table is calculated.

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  • Since p is determined experimentally and tabulated as a function of v, the velocity is taken as the argument of the ballistic table; and taking Av =10, the average value of p in the interval is used to determine AT.

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  • Denoting the value of T at any velocity v by T (v), then (8) T(v) = sum of all the preceding values of AT plus an arbitrary constant, expressed by the notation (9) T(v) =Z(Av)/gp+ a constant, or fdv/gp+ a constant, in which p is supposed known as a function of v.

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  • The constant may be any arbitrary number, as in using the table the difference only is required of two tabular values for an initial velocity V and final velocity v; and thus (to) T(V) - T(v) = Ev Ov/gp or fvdv/gp; and for a shot whose ballistic coefficient is C (II) t=C[T(V) - T(v)].

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  • Denoting by S(v) the sum of all the values of AS up to any assigned velocity v, (is) S(v) =E(OS)+ a constant, by which S(v) is calculated from AS, and then between two assigned velocities V and v, V AT, = vAv or rvvdv vgp gp' and if s feet is the advance of a shot whose ballistic coefficient is C, (17) s=C[S(V) - S(v)].

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  • In an extended table of S, the value is interpolated for unit increment of velocity.

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  • These functions, T, S, D, 1, A, are shown numerically in the following extract from an abridged ballistic table, in which the velocity is taken as the argument and proceeds by an increment of 10 f/s; the column for p is the one determined by experiment, and the remaining columns follow by calculation in the manner explained above.

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  • In any region of velocity where it is possible to represent p with sufficient accuracy by an empirical formula composed of a single power of v, say v m, the integration can be effected which replaces the summation in (to), (16), and (24); and from an analysis of the Krupp experiments Colonel Zabudski found the most appropriate index m in a region of velocity as given in the following table, and the corresponding value of gp, denoted by f (v)or v m lk or its equivalent Cr, where r is the retardation.

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  • Given the ballistic coefficient C, the initial velocity V, and a range of R yds.

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  • The last column in the Range Table giving the inches of penetration into wrought iron is calculated from the remaining velocity by an empirical formula, as explained in the article Armour Plates.

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  • Also the velocity v at the end of the arc is given by (87) ve = u e sec 0 cos n.

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  • In this table (93) sin 20=Ca, where a is a function tabulated for the two arguments, V the initial velocity, and R/C the reduced range in yards.

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  • This muzzle velocity is about 5% greater than the 2150 f/s of the range table, so on these considerations we may suppose about 10% of work is lost by friction in the bore; this is expressed by saying that the factor of effect is f =0.9.

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  • According to the wave-theory of light, refraction is due to a change of velocity when light passes from one medium to another.

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  • The phenomenon of dispersion shows that in dispersive media the velocity is different for lights of different wave-lengths.

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  • In free space, light of all wave-lengths is propagated with the same velocity, as is shown by the fact that stars, when occulted by the moon or planets, preserve their white colour up to the last moment of disappearance, which would not be the case if one colour reached the eye later than another.

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  • It can be shown mathematically that the velocity of propagation will be greatly increased if the frequency of the light-wave is slightly greater, and greatly diminished if it is slightly less than the natural frequency of the molecules; also that these effects become less and less marked as the difference in the two frequencies increases.

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  • The great river receives an abundant load of silt from its tributaries, and takes up ano lays down silt from its own bed and banks with every change of velocity.

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  • The load of silt borne down stream by the river finally, after many halts on the way, reaches the waters of the Gulf, where the decrease of velocity, aided by the salinity of the sea water, causes the formation of a remarkable delta, leaving less aggraded areas as shallow lakes (Lake Pontchartrain on the east, and Grand Lake on the west of the river).

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  • At Canyon City it passes out of the Rockies through the Grand Canyon of the Arkansas; then turning eastward, and soon a turbid, shallow stream, depositing its mountain detritus, it flows with steadily lessening gradient and velocity in a broad, meandering bed across the prairies and lowlands of eastern Colorado, Kansas, Oklahoma and Arkansas, shifting its direction sharply to the south-east in central Kansas.

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  • The great velocity of electrical transmission suggested the possibility of utilizing it for sending messages; and, after many experiments and the practical advice and business-like co-operation of William Fothergill Cooke (1806-1879), a patent for an electric telegraph was taken out in their joint names in 1837.

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  • The tidal currents, or races, or roost (as some of them are called locally, from the Icelandic) off many of the isles run with enormous velocity, and whirlpools are of frequent occurrence, and strong enough at times to prove a source of danger to small craft.

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  • Since the capacity of a stream to carry matter in suspension is proportional to its velocity, it follows that any circumstance tending to retard the rate of flow will induce deposition.

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  • Thus a fall in the gradient at any point in the course of a stream; any snag, projection or dam, impeding the current; the reduced velocity caused by the overflowing of streams in flood and the dissipation of their energy where they enter a lake or the sea, are all contributing causes to alluviation, or the deposition of streamborne sediment.

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  • Reinders (Ber., 1896, 29, p. 1369), who found that the reaction is monomolecular, and that the velocity constant of the reaction is proportional to the amount of the hydrochloride of the base present and also to the temperature, but is independent of the concentration of the diazoamine.

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  • This apparent motion is due to the finite velocity of light, and the progressive motion of the observer with the earth, as it performs its yearly course about the sun.

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  • If the bearer be stationary, rain-drops will traverse the tube without touching its sides; if, however, the person be walking, the tube must be inclined at an angle varying as his velocity in order that the rain may traverse the tube centrally.

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  • Bradley recognized the fact that the experimental determination of the aberration constant gave the ratio of the velocities of light and of the earth; hence, if the velocity of the earth be known, the velocity of light is determined.

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  • If N be the frequency of a homogeneous vibration sent out by a molecule at rest, the apparent frequency will be N (1 v/ V), where V is the velocity of light and v is the velocity of the line of sight, taken as positive if the distance from the observer increases.

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  • If all molecules moved with the velocity of mean square, the line would be drawn out into a band having on the frequency scale a width 2Nv/V, where v is now the velocity of mean square.

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  • If the motion were that of a body at white heat, or say a temperature of loco, the velocity of mean square would be 39co metres per second and the apparent width of the band would be doubled.

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  • Hemsalech 1 have measured the velocity with which the luminous molecules are projected from metallic poles when a strong spark is passed through the air interval which separates the poles.

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  • In the case of some metals, notably bismuth, the velocity measured was different for different lines, which seems intelligible only on the supposition that the metal vapour consists of different vibrating systems which can differ with different velocities.

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  • The "mean moon" is a fictitious moon which moves around the earth with a uniform velocity and in the same time as the real moon.

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  • In order to exert force, or at all events that force of reciprocal pressure which we best understand, and on which, in impact, the third law of motion was founded, there are always at least two bodies, enduring, triply extended, mobile, each inert, mutually impenetrable or resistent, different yet similar; and in order to have produced any effect but equilibrium, some bodies must at some time have differed either in mass or in velocity, otherwise forces would only have neutralized one another.

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  • It shows that the bodies impress on one another opposite changes of velocity inversely as their weights or masses; and that in doing so they always begin by reducing one another to a joint mass with a common velocity, whatever they may do afterwards in consequence of their elasticities.

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  • The two bodies therefore do not penetrate one another, but begin by acting on one another with a force precisely sufficient, instead of penetrating one another, to cause them to form a joint mass with a common velocity.

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  • Bodies then are triply extended substances, each occupying enough space to prevent mutual penetration, and by this force of mutual impenetrability or interresistance cause one another to form a joint mass with a common velocity whenever they collide.

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  • Withdraw this foundation of bodies as inter-resisting forces causing one another in collision to form a joint mass with a common velocity but without penetration, and the evidence of the third law disappears; for in the case of attractive forces we know nothing of their modus operandi except by the analogy of the collision of inter-resisting bodies, which makes us believe that something similar, we know not what, takes place in gravity, magnetism, electricity, &c. Now, Mach, though he occasionally drops hints that the discovery of the law of collision comes first, yet never explains the process of development from it to the third law of motion.

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  • He has therefore lost sight of the truths that bodies are triply extended, mutually impenetrable substances, and by this force causes which reduce one another to a joint mass with a common velocity on collision, as for instance in the ballistic pendulum; that these forces are the ones we best understand; and that they are reciprocal causes of the common velocity of their joint mass, whatever happens afterwards.

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  • The chief results we have found against idealism are that bodies have not been successfully analysed except into bodies, as real matter; and that bodies are known to exert reciprocal pressure in reducing one another to a joint mass with a common velocity by being mutually impenetrable, as real forces.

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  • Galileo proceeded to measure the motion of a body on a smooth, fixed, inclined plane, and found that the law of constant acceleration along the line of slope of the plane still held, the acceleration decreasing in magnitude as the angle of inclination was reduced; and he inferred that a body, moving on a smooth horizontal plane, would move with uniform velocity in a straight line if the resistance of the air, and friction due to contact with the plane, could be eliminated.

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  • Such statements as that a body moves in a straight line, and that it has a certain velocity, have no meaning unless the base, relative to which the motion is to be reckoned, is defined.

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  • Newton assumed the possibility of choosing a base such that, relatively to it, the motion of any particle would have only such divergence from uniform velocity in a straight line as could be expressed by laws of acceleration dependent on its relation to other bodies.

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  • Suppose two small smooth spherical bodies which can be regarded as particles to be brought into collision, so that the velocity of each, relative to any base which is unaffected by the collision, is suddenly changed.

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  • The additions of velocity which the two bodies receive respectively, relative to such a base, are in opposite directions, and if the bodies are alike their magnitudes are equal.

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  • If the bodies though of the same substance are of different sizes, the magnitudes of the additions of velocity are found to be inversely proportional to the volumes of the bodies.

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  • In fact, experiments upon the changes of velocity of bodies, due to a mutual influence between them, bring to light a property of bodies which may be specified by a quantity proportional to their volumes in the case of bodies which are perceived by other tests to be of one homogeneous substance, but otherwise involving also another factor.

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  • When, as in the case of contact, a mutual relation is perceived between the motions of two particles, the changes of velocity are in opposite directions, and the ratio of their magnitudes determines the ratio of the masses of the particles; the motion being reckoned relative to any base which is unaffected by the change.

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  • This test involves only changes of velocity, and so does not distinguish between two bases, each of which moves relatively to the other with uniform velocity without rotation.

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  • Hence the force required to drive one gramme-molecule of sugar through water with a velocity of one centimetre per second may be calculated as some thousands of millions of kilogrammes weight.

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  • The resistance offered by the liquid, and therefore the force F, required to drive one grammemolecule through the liquid with unit velocity is the sum of the corresponding quantities for the individual ions.

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  • This is precisely the number found from the velocity of sound in argon as determined by Kundt's method, and it leaves no room for any sensible energy of rotatory or vibrational motion.

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  • The electromotive force is practically constant no matter what the velocity of the disks, but according to some observers the internal resistance decreases as the velocity increases.

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  • The mean velocity of their flow seldom exceeds 4.9 ft., but rises to 6.4 ft.

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  • In the lower reaches of the streams the velocity and slope are of course affected by the tides.

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  • The reason why the frictional resistance would be further increased is the very simple one that the increase in the rate of production implies directly a corresponding increase in the quantity of blast forced through, and hence in the velocity of the rising gases, because the chemical work of the blast furnace needs a certain quantity of blast for each ton of iron made.

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  • In short, to increase the rate of production by lengthening the furnace increases the frictional resistance of the rising gases, both by increasing their quantity and hence their velocity and by lengthening their path.

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  • We see how powerful must be the lifting effect of the rising gases when we reflect that their velocity in a too ft.

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  • Conceive these gases passing at this great velocity through the narrow openings between the adjoining lumps of coke and ore.

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  • Indeed, the velocity must be far greater than this where the edge or corner of one lump touches the side of another, and the only room for the passage of this enormous quantity of gas is that left by the roughness and irregularity of the individual lumps.

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  • If the linear velocity of the cups in feet a second is V 1, and the linear velocity of the jet is V2, then the velocity of the jet relative to the cup is V2 - V1 feet a second, and if the whole energy of the water is to be given up to the cups, the water must leave the cup with zero absolute velocity.

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  • But its velocity relative to the cup, as it passes backwards, is - (V 2 - V 1), and since the forward velocity of the cup is Vi, the absolute velocity of the water is - (V2 - Vi) +VI or2V i - V2.

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  • This will become zero if V 1 is 2V 2, that is, if the linear velocity of the cupcentres is one-half that of the jet of water impinging upon them.

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  • The tides, which are very high, run into it with amazing velocity, but at low water the bottom is left nearly dry for some distance below the latitude of the town of Cambay.

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  • The velocity of propagation of temperature waves will be the same under similar conditions in two substances which possess the same diffusivity, although they may differ in conductivity.

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  • Uniformity of temperature could only be secured by using a high velocity of flow, or violent stirring.

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  • For instance, the velocity of propagation of a wave having a period of a day is nearly twenty times as great as that of a wave with a period of one year; but on the other hand the penetration of the diurnal wave is nearly twenty times less, and the shorter waves die out more rapidly.

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  • The magnitude of the stress per unit area parallel to the direction of flow is evidently proportional to the velocity gradient, or the rate of change of velocity per cm.

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  • If the effects depended merely on the velocity of translation of the molecules, both conductivity and viscosity should increase directly as the square root of the absolute temperature; but the mean free path also varies in a manner which cannot be predicted by theory and which appears to be different for different gases (Rayleigh, Proc. R.S., January 1896).

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  • An auroral curtain travelling with considerable velocity would approach from the south, pass right overhead and retire to the north.

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  • According to numerous observations made at Cape Thorsden, the apparent angular velocity of arcs increases on the average with their altitude.

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  • The velocity 109.09 was much the largest observed, the next being 52.38; both were from observations lasting under half a minute.

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  • There is also difficulty in ensuring that the observations shall be simultaneous, an important matter especially when the apparent velocity is considerable.

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  • Cathode rays usually have a velocity about a tenth that of light, but in exceptional cases it may approach a third of that of light.

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  • Hertzian waves have the velocity of light itself.

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  • Now if a be the amplitude expressed in millimetres, and t the period expressed in seconds, then the maximum velocity of an earth particle as it vibrates to and fro equals 27ra/t, whilst the maximum acceleration equals 4,r 2 0 2.

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  • For example, if a body, say a coping-stone, has been thrown horizontally through a distance a, and fallen from a height b, the maximum horizontal velocity with which it was projected equals !

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  • Another remarkable phenomenon is the zobaa, a lofty whirlwind of sand resembling a pillar, which moves with great velocity.

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  • The mean velocity of winds for 1906 was 110 m.; the maximum recorded being 148 in May, the minimum velocity recorded being 76 in December.

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  • The optical apparatus generally consists of a mirror mounted on an axis parallel to the axis of the earth, and rotated with the same angular velocity as the sun.

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  • It is easily seen that if the mirror be rotated at the same angular velocity as the sun the right ascensions will remain equal throughout the day, and therefore this device reflects the rays in the direction of the earth's axis; a second fixed mirror reflects them in any other fixed direction.

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  • By adjusting the right ascension of the plane ABC and rotating the axis with the angular velocity of the sun, it follows that BC will be the direction of the solar rays throughout the day.

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  • By the use of a revolving mirror similar to that used by Sir Charles Wheatstone for measuring the rapidity of electric currents, he was enabled in 1850 to demonstrate the greater velocity of light in air than in water, and to establish that the velocity of light in different media is inversely as the refractive indices of the media.

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  • With Wheatstone's revolving mirror he in 1862 determined the absolute velocity of light to be 298,000 kilometres (about 185,000 m.) a second, or 10,000 kilom.

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  • Further, by causing the hour circle, and with it the polar axis, to rotate by clockwork or some equivalent mechanical contrivance, at the same angular velocity as the earth on its axis, but in the opposite direction, the telescope will, apart from the effects of refraction, automatically follow a star from rising to setting.

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  • The Direct Methods Of Measuring The Ratio S/S, By The Velocity Of Sound And By Adiabatic Expansion, Are Sufficiently Described In Many Text Books.

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  • The suggestion was made, and seems to be the true explanation, that what was actually witnessed was the wave of light due to the outburst of the nova, spreading outwards with its velocity of 186,000 m.

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  • By means of the spectroscope it is possible to determine the relative orbital velocity of the two components, and this when compared with the period fixes the absolute dimensions of the orbit; the apparent dimensions of the orbit being known from visual observations the distance can then be found.

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    0
  • The velocity in the line of sight can be determined by spectroscopic observation, so that in a few cases the motion of the star is completely known.

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    0
  • Probably the velocity of Arcturus is also over 100 m.

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    0
  • Campbell the average velocity in space of a star is 21.2 m.

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    0
  • Regarded as a linear velocity, the parallactic motion is the same for all stars, being exactly equal and opposite to the solar motion; but its amount, as measured by the corresponding angular displacement of the star, is inversely proportional to the distance of the star from the earth, and foreshortening causes it to vary as the sine of the angular distance from the apex.

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    0
  • Campbell from the radial motions of 280 stars found the velocity to be 20 kilometres per second with a probable error of 12 km.

    0
    0
  • Halm deduced a velocity of 20.8 km.

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    0
  • In 1728 was published "A Letter from Dr Clarke to Benjamin Hoadly, F.R.S., occasioned by the controversy relating to the Proportion of Velocity and Force in Bodies in Motion," printed in the Philosophical Transactions.

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    0
  • At Vienna he had lessons in pianoforte playing from Carl Czerny of " Velocity " fame, and from Salieri in harmony and analysis of scores.

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    0
  • This is called the curve of positions or space-time curve; its gradient represents the velocity.

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    0
  • It is a matter of ordinary observation that different bodies acted on by the same force, or what is judged to be the same force, undergo different changes of velocity in equal times.

    0
    0
  • The product mu of the mass into the velocity is called the momentum or (in Newtons phrase) the quantity of motion.

    0
    0
  • On the Newtonian system the motion of a particle entirely uninfluenced by other bodies, when referred to a suitable base, would be rectilinear, with constant velocity.

    0
    0
  • If we take as rough values a=21 X,o6 feet, g=32 foot-second units, we get a velocity of 36,500 feet, or about seven miles, per second.

    0
    0
  • We may briefly notice the case of resistance varying as the square of the velocity, which is mathematically simple.

    0
    0
  • Tait that a similar representation of the type (30) is obtained if we replace the circle by an equiangular spiral described, with a constant angular velocity about the pole, in the direction of diminishing radius vector.

    0
    0
  • For purposes of mathematical treatment a force which produces a finite change of velocity in a time too short to be appreciated is regarded as infinitely great, and the time of action as infinitely short.

    0
    0
  • Thus the unit of velocity is that of a point describing the unit of length in the unit of time; it may be denoted by LTi, this symbol indicating that the magnitude of the unit in question varies directly as the unit of length and inversely as the unit of time.

    0
    0
  • The unit of acceleration is the acceleration of a point which gains unit velocity in unit time; it is accordingly denoted by LT2.

    0
    0
  • As 6t is indefinitely diminished, the vector OU will tend to a definite limit OV; this is adopted as the definitiov of the velocity of the moving point at the instant t.

    0
    0
  • The momentum of a particle is the vector obtained by multiplying the velocity by the mass in.

    0
    0
  • In symbols, if v be the velocity and p the perpendicular from 0 to the tangent to the path, pv=h, (1)

    0
    0
  • Hence the character of the orbit depends on whether the velocity at any point is, less than, equal to, or greater than the velocity from infinity, as it is called.

    0
    0
  • In order that the spiral may be described it is necessary that the velocity of projection should be adjusted to make h=iju.

    0
    0
  • If A or B vanish we have an equiangular spiral, and the velocity at infinity is zero.

    0
    0
  • A point on a central orbit where the radial velocity (drfdt) vanishes is called an apse, and the corresponding radius is called an apse-line.

    0
    0
  • If the force is always the same at the same distance any apse-line will divide the orbit symmetrically, as is seen by imagining the velocity at the apse to be reversed.

    0
    0
  • If in a central orbit the velocity is equal to the velocity from infinity, we have, from (5),

    0
    0
  • The question presents itself whether ther then is any other law of force, giving a finite velocity from infinity, under which all finite orbits are necessarily closed curves.

    0
    0
  • At the beginning of 13 the velocity of a moving point P was represented by a vector OV drawn from a fixed origin 0.

    0
    0
  • The locus of the point V is called the hodograp/z (q.v.); and it appears that the velocity of the point V along the hodograph represents in magnitude and in directon tbt acceleration in the original orbit.

    0
    0
  • In the motion of a projectile under gravity the hodograph is a vertical line described with constant velocity.

    0
    0
  • In elliptic harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the hodograph is therefore an ellipse similar to the actual orbit.

    0
    0
  • In the case of a particle oscillating under gravity on a smooth cycloid from rest at the cusp the hotlograph is a circle through the pole, described with constant velocity.

    0
    0
  • For example, the mass-centre of I system free from extraneous force will describe a straight lin with constant velocity.

    0
    0
  • Then dU/dt, =w say, is the angular velocity of the body.

    0
    0
  • The angular velocity being constant, the effective force on a particle m at a distance r from Oz is snw2r toward& this axis, and its components are accordingly w2mx, wfmy, 0.

    0
    0
  • If the extraneous forces have zero moment about G the angular velocity 0 is constant.

    0
    0
  • The circle is described with the constant angular velocity o.

    0
    0
  • As an example of this latter type, suppose that a sphere is placed on the highest point of a fixed sphere and set spinning about the vertical diameter with the angular velocity n; it will appear that under a certain condition the motion of G consequent on a slight disturbance will be oscillatory.

    0
    0
  • Now T = 3/41w1, where w is the angular velocity and I is the moment of inertia about the instantaneous axis.

    0
    0
  • The motion of the body relative to 0 is therefore completely represented if we imagine the momental ellipsoid at 0 to roll without sliding on a plane fixed in space, with an angular velocity proportional at each instant to the radius-vector of the point of contact.

    0
    0
  • The angular velocity (r) about this axis is then constant.

    0
    0
  • As a first application of the equations (2) take the case of a solid constrained to rotate with constant angular velocity to about a fixed axis (1, m, n).

    0
    0
  • The physical characteristics of a normal mode are that an impulse of a particular normal type generates an initial velocity of that type only, and that a constant extraneous force of a particular normal type maintains a displacement of that type only.

    0
    0
  • It consists of two elements, the velocity ratio, which is the ratio of any two magnitudes bearing to each other the proportions of the respective velocities of the two points at a given instant, and the directional relation, which is the relation borne to each other by the respective directions of the motions of the two points at the same given instant.

    0
    0
  • The comparative motion of two points at a given instant is capable of being completely expressed by one of Sir William Hamiltons Quaternions,the tensor expressing the velocity ratio, and the versor the directional relation.

    0
    0
  • Let a represent the area of the section of a piston made by a plane perpendicular to its direction of motion, and v its velocity, which is to be considered as positive when outward, and negative when inward.

    0
    0
  • Let the angular velocity of the rotation be denoted by a=dO/dt, then the linear velocity of any point A at the distance r from the axis is or; and the path of that point is a circle of the radius r described about the axis.

    0
    0
  • This is the principle of the modification of motion by the lever, which consists of a rigid body turning about a fixed axis called a fulcrum, and having two points at the same or different distances from that axis, and in the same or different directions, one of which receives motion and the other transmits motion, modified in direction and velocity according to the above law.

    0
    0
  • Velocity Ratio of Components of Motion.As the distance between any two points in a rigid body is invariable, the projections of their velocities upon the line joining them must be equal.

    0
    0
  • The line T on the surface bbb has for the instant no velocity it a direction perpendicular to AB; becau2e for the instant it touches, without sliding, the line T on the fixed surface aaa.

    0
    0
  • The line T on the surface bbb has also for the instant no velocity in the plane AB; for it has just ceased to move towards the fixed surface aaa, and is just about to begin to move away from that surface.

    0
    0
  • The velocity of any point in the axis of figure B is va=y.TB; (4)

    0
    0
  • Let -y denote the total angular velocity of the rotation of the cone B about the instantaneous axis, $ its angular velocity about the axis OB relatively to the plane AOB, and a the angular velocity with which the plane AOB turns round the axis OA.

    0
    0
  • Let yr be the linear velocity of the point E fixed in the plane of axes AOB.

    0
    0
  • Now, as the line of contact OT is for the instant at rest on the rolling cone as well as on the fixed cone, the linear velocity of the point E fixed to the plane AOB relatively to the rolling cone is the same with its velocity relatively to the fixed cone.

    0
    0
  • Then the motion of P is perpendicular to the plane OPQ, and its velocity is v,.= y.

    0
    0
  • Let V5 denote the velocity of advance at a given instant, which of course is common to all the particles of the body; a the angular velocity of the rotation at the same instant; 2,r = 6.2832 nearly, the circumference of a circle of the radius unity.

    0
    0
  • The ratio of the two components of that velocity is = p/2lrr = tan 0.

    0
    0
  • In the investigation, therefore, of the comparative motion, of the driver and follower, in an elementary combination, it is unnecessary to consider relations of angular direction, which are already fixed by the connection of each piece with the frame; so that the inquiry is confined to the determination of the velocity ratio, and of tbe directional relation, so far only as it expresses the connection between forward and backward movements of the driver and follower.

    0
    0
  • The line of action or of connection of the driver and follower is a line traversing a pair of points in the driver and follower respectively, which are so connected that the component of their velocity relatively to each other, resolved along the line of connection, is null.

    0
    0
  • Williss classification is founded, in the first place, on comparative motion, as expressed by velocity ratio and directional relation, and in the second place, on the mode of connection of the driver and follower.

    0
    0
  • That the linear velocity of a shifting piece in rolling contact with a turning piece is equal to the product of the angular velocity of the turning piece by the perpendicular distance from its axis to a pair of points of contact.

    0
    0
  • If the velocity ratio is to be -constant, as in, fYi Williss Class A, the wheels must be circular; and this is the most common form for wheels.

    0
    0
  • If the velocity ratio is to be variable, as in s Williss Class B, the figures of the wheels are a pair of rolling curves, subject to the condition that the distance between their poles (which are the centres of rotation) shall be constant.

    0
    0
  • When the velocity ratio is variable, the line of contact will shift its position in the plane C1OC2, and the wheels will be cones, with eccentric or irregular bases.

    0
    0
  • In every case which occurs in practice, however, the velocity ratio is FIG.

    0
    0
  • Hence also, in any pair of circular wheels which rotate continuously for one revolution or more, the ratio of the numbers of teeth and its reciprocal the angular velocity ratio must be expressible in whole numbers.

    0
    0
  • They there fore study that the numbers of teeth in each pair of wheels whici work together shall either be prime to each other, or shall hav their greatest common divisor as small as is consistent with velocity ratio suited for the purposes of the machine.

    0
    0
  • The angular velocity ratio due to the sliding contact of the teeth will be the same with that due to the rolling contact of the pitch-circles, if the line of connection of the teeth cuts the Ca line of centres at the pitchpoint.

    0
    0
  • Thus the relative motion of the wheels is unchanged; but I is considered as fixed, and 2 has the total motion, that is, a rotation about the instantaneous axis I, with the angular velocity cii+a1.

    0
    0
  • Hence the velocity of sliding is that due to this rotation about I, with the radius IT; that is to say, its value is (ai+ai).IT; (26)

    0
    0
  • Any other convenient figure may be assumed for the path of contact, and the corresponding forms of the teeth found by determining what curves a point T, moving along the assumed path of contact, will trace on two disks rotating round the centres of the wheels with angular velocities bearing that relation to the component velocity of T along TI, which is given by Principle II.

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  • Coupling of Parallel AxesOldhams CouplingA coupling is a mode of connecting a pair of shafts so that they shall rotate in the same direction with the same mean angular velocity.

    0
    0
  • Pulleys and drums for communi cating a constant velocity ratio are circular.

    0
    0
  • The speed-cones are either continuous cones or conoids, as A, B, whose velocity ratio can be varied gradually while they are in motion by shifting the belt, or sets of pulleys whose radii vary by steps, as C, D, in which case the velocity ratio can be changed by shifting the belt from one pair of pulleys to another.

    0
    0
  • The axes of rotation of a pair of turning pieces connected by a link are almost always parallel, and perpendicular to the line of connection n which case the angular velocity ratio at any instant is the recipocal of the ratio of the common perpendiculars let fall from the me of connection upon the respective axes of rotation.

    0
    0
  • The velocity of the other connected point at such an instant is null, unless it also reaches a dead-point at the same instant, so that the line of connection is in the plane of the two axes of rotation, in which case the velocity ratio is indeterminate.

    0
    0
  • Let ci denote the velocity 1sf Tf at any given instant; v2 that of T,.

    0
    0
  • The comparative motion of the first driver and last follower is obtained by combining the proportions expressing by their terms the velocity ratios and by their signs the directional relations of the several elementary combinations of which the train consists.

    0
    0
  • It is often a question of importance to determine the number of teeth in a train of wheels best suited for giving a determinate velocity ratio to two axes.

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  • It was shown by Young that, to do this with the least total number of teeth, the velocity ratio of each elementary combination should approximate as nearly as possible to 3.59., This would in many cases give too many axes; and, as a useful practical rule, it may be laid down that from 3 to 6 ought to be the limit of the velocity ratio of an elementary combination in wheelwork.

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  • Let B/C be the velocity ratio required, reduced to its least terms, and let B be greater than C. If B/C is not greater than 6, and C lies between the prescribed minimum number of teeth (which may be called t) and its double 2t, then one pair of wheels will answer the purpose, and B and C will themselves be the numbers required.

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  • Then, if possible, B and, C themselves are to be resolved each into rnI factors (counting 1 as a factor), which factors, or multiples of them, shall be not less than t nor greater than 6t; or if B and C contain inconveniently large prime factors, an approximate velocity ratio, found by the method of continued fractions, is to be substituted for B/C as before.

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  • Double Hookes Coupling.It has been shown in 66 that the velocity ratio of a pair of shafts coupled by a universal joint fluctuates between the limits cos 0 and 1/cos 0.

    0
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  • Then, from the principles of 60 it is evident that at each instant ai/ai = ai/aa, and consequently that ai; so that the fluctuations of angular velocity ratio caused by the first coupling are exactly neutralized by the second, and the first and last shafts have equal angular velocities at each instant.

    0
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  • Required the relation between the velocity of translation 02 of W and the angular velocity af of the differential barrel.

    0
    0
  • The velocity of the point F is ai.

    0
    0
  • The velocity of the point G is ai.

    0
    0
  • If the cord be fixed to the framework at the point B, instead of being wound on a barrel, the velocity of W is half that of AF.

    0
    0
  • The velocity of advance of B relatively to C is (according to 32) aPi, and of A relatively to B (according to 57) aPi; hence the velocity of A relatively to C is a(piPi), (46)

    0
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  • Method 1.By reference to 30 it will be seen that the motion of a cylinder rolling on a fixed cylinder is one of rotation about an instantaneous axis T, and that the velocity both as regards direction and magnitude is the same as if the rolling piece B were for the instant turning about a fixed axis coincident with the instantaneous axis.

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  • There is no restriction on the shape of these rolling axodes; they may have any shape consistent with i,olling (that is, no slipping is permitted), and the relative velocity of a point P is still found by considering it with regard to the instantaneous centre.

    0
    0
  • Method 2.The second method is based upon the vector representation of velocity, and may be illustrated by applying it to the four-bar chain.

    0
    0
  • Hence set out the possible direction of Bs motion in the velocity diagram, namely cbi, at right angles to CB.

    0
    0
  • Hence draw a line through O in the velocity diagram at right angles to AB to cut cbi in b.

    0
    0
  • Then Ob is the velocity of the point b in magnitude and direction, and cb is the tangential velocity of B relatively to C. Moreover, whatever be the actual magnitudes of the velocities, the instantaneous velocity ratio of the points C and B is given by the ratio Oc/Ob.

    0
    0
  • It will be understood that there is a new velocity diagram for every new configuration of the mechanism, and that in each new diagram the image of the rod will be different in scale.

    0
    0
  • Following the method indicated above for a kinematic chain in general, there will be obtained a velocity diagram similar to that of fig.

    0
    0
  • The lines joining the ends of these several velocities are the several tangential velocities, each being the velocity image of a link in the chain.

    0
    0
  • These several images are not to the same scale, so that although the images may be considered to form collectively an image of the chain itself, the several members of this chain-image are to different scales in any one velocity diagram, and thus the chainimage is distorted from the actual proportions of the mechanism which it represents.

    0
    0
  • That is to say, although the instantaneous centre is a centre of no velocity for the instant, it is not a centre of no acceleration, and in fact the centre of no acceleration is in general a quite different point.

    0
    0
  • The magnitude of the radial acceleration is given by the expression vi/BC, v being the velocity of the point B about the point C. This velocity can always be found from the velocity diagram of the chain of which the link forms a part.

    0
    0
  • In applying this principle to the drawing of an acceleration diagram for a mechanism, the velocity diagram of the mechanism must be first drawn in order to afford the means of calculating the several radial accelerations of the links.

    0
    0
  • Examples, completely worked out, of velocity and acceleration diagrams for the slider crank chain, the four-bar chain, and the mechanism of the Joy valve gear will be found in ch.

    0
    0
  • Removing the summation signs in equation (52) in order to restrict its application to two points and dividing by the common time interval during which the respective small displacements ds and ds were made, it becomes Pdsfdt = Rds/dt, that is, Pv = Rv, which shows that the force ratio is the inverse of the velocity ratio.

    0
    0
  • It follows at once that any method which may be available for the determination of the velocity ratio is equally available for the determination of the force ratio, it being clearly understood that the forces involved are the components of the actual forces resolved in the direction of motion of the points.

    0
    0
  • These velocity ratios are known by the construction of the mechanism, and are independent of the absolute speed.

    0
    0
  • Moment of Friction.The work performed in a unit of time in overcoming the friction of a pair of surfaces is the product of the friction by the velocity of sliding of the surfaces over each other, if that is the same throughout the whole extent of the rubbing surfaces.

    0
    0
  • If that velocity is different for different portions of the rubbing surfaces, the velocity of each portion is to be multiplied by the friction of that portion, and the results summed or integrated.

    0
    0
  • When the relative motion of the rubbing surfaces is one of rotation, the work of friction in a unit of time, for a portion of the rubbing surfaces at a given distance from the axis of rotation, may be found by multiplying together the friction of that portion, its distance from the axis, and the angular velocity.

    0
    0
  • In 45 the velocity of sliding at any instant has been given, viz.

    0
    0
  • Let v be the common velocity of the two pitch-circles, ri, C2, their radii; then the above equation becomes /1 I

    0
    0
  • The equations 65 and 66 are applicable to a kind of brake called a friction-strap, used to stop or moderate the velocity of machines by being tightened round a pulley.

    0
    0
  • Friction-CouplingsFriction is useful as a means of communicating motion where sudden changes either of force or velocity take place, because, being limited in amount, it may be so adjusted as to limit the forces which strain the pieces of the mechanism within the bounds of safety.

    0
    0
  • Its moment is found by multiplying the normal pressure between the rolling surfaces by an arm, whose length depends on the nature of the rolling surfaces, and the work lost in a unit of time in overcoming it is the product of its moment by the angular velocity of the rolling surfaces relatively to each other.

    0
    0
  • Neglecting the mass of the shaft itself, when the shaft rotates with an angular velocity a, the centrifugal force Wae/g will act upon the shaft and cause its axis to deflect from the axis of rotation a distance, y say.

    0
    0
  • In finding the position in which the bob will revolve with a given angular velocity, a, for most practical cases connected with machinery the mass of the rod may be considered as insensible compared with that of the bob.

    0
    0
  • Working of Machines of Varying Velocity.

    0
    0
  • General Principles.In order that the velocity of every piece of a machine may be uniform, it is necessary that the forces acting on each piece should be always exactly balanced.

    0
    0
  • Actual Energy of a Shifting Body.The energy which must be exerted on a body of the weight w, to accelerate it from a state of rest up to a given velocity of translation v, and the equal amount of work which that body is capable of performing by overcoming resistance while being retarded from the same velocity of translation v to a state of rest, is wvfI2g.

    0
    0
  • The energy stored or restored, as the case may be, by the deviations of velocity of a body or a system of bodies, is the amount by which the actual energy is increased or diminished.

    0
    0
  • Principle of the Conservation of Energy in Machines.The following principle, expressing the general law of the action of machines with a velocity uniform or varying, includes the law of the equality of energy and work stated in 89 for machines of uniform speed.

    0
    0
  • Let a small body of the weight w undergo translation in a circulai path of the radius p, with the angular velocity of deflexion a, so that the common linear velocity of all its particles is v=ap. Then the actual energy of that body is WV2/2g = Waip2/2g.

    0
    0
  • The product wp/g, by which the half-square of the angular velocity is multiplied, is called the moment of inertia of the revolving body.

    0
    0
  • Flywheels.A flywheel is a rotating piece in a machine, generally shaped like a wheel (that is to say, consisting of a rim with spokes), and suited to store and restore energy by the periodical variations in its angular velocity.

    0
    0
  • The principles according to which variations of angular velocity store and restore energy are the same as those of 117, only substituting moment of inertia for mass, and angular for linear velocity.

    0
    0
  • The actual energy due to the rotation of the fly, with its mean angular velocity, is equal to one-half of the periodical excess of energy multiplied by the steadiness.

    0
    0
  • Brakes.A brake is an apparatus for stopping and diminishing the velocity of a machine by friction, such as the friction-strap already referred to in 103.

    0
    0
  • Let da be the deviation of angular velocity to be produced in the interval dt, and I the moment of the inertia of the body about an axis through its centre of gravity; then 1/8Id(&) = Iada is the variation of the bodys actual energy.

    0
    0
  • When the link forms part of a mechanism the respective accelerations of two points in the link can be determined by means of the velocity and acceleration diagrams described in 82, it being understood that the motion of one link in the mechanism is prescribed, for instance, in the steam-engines mechanism that the crank shall revolve uniformly.

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    0
  • One of the most important investigations he made in this way was to find out, as he expressed it, " what power of the velocity the resistance is proportional to."

    0
    0
  • Cavendish meant by the term " velocity " what we now call the current, and by " resistance " the electromotive force which maintains the current.

    0
    0
  • Coupling together these ideas he was finally enabled to prove that the propagation of electric and magnetic force takes place through space with a certain velocity determined by the dielectric constant and the magnetic permeability of the medium.

    0
    0
  • If we imagine the current in the conductor to be instantaneously reversed in direction, the magnetic force surrounding it would not be instantly reversed everywhere in direction, but the reversal would be propagated outwards through space with a certain velocity which Maxwell showed was inversely as the square root of the product of the magnetic permeability and the dielectric constant or specific inductive capacity of the medium.

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    0
  • Maxwell showed in this paper that the velocity of propagation of an electromagnetic impulse through space could also be determined by certain experimental methods which consisted in measuring the same electric quantity, capacity, resistance or potential in two ways.

    0
    0
  • Maxwell suggested new methods for the determination of this ratio of the electrostatic to the electromagnetic units, and by experiments of great ingenuity was able to show that this ratio, which is also that of the velocity of the propagation of an electromagnetic impulse through space, is identical with that of light.

    0
    0
  • Experimental methods were devised for the further exact measurements of the electromagnetic velocity and numerous determinations of the dielectric constants of various solids, liquids and gases, and comparisons of these with the corresponding optical refractive indices were conducted.

    0
    0
  • Abraham, showed it to be very close to the best determinations of the velocity of light (see Physical Units).

    0
    0
  • These effects, as Hertz showed, indicated the establishment of stationary electric waves in space and the propagation of electric and magnetic force through space with a finite velocity.

    0
    0
  • Electric waves are produced wherever electrons are accelerated or retarded, that is, whenever the velocity of an electron is changed or accelerated positively or negatively.

    0
    0
  • He studied the nature of muscular contraction, causing a muscle to record its movements on a smoked glass plate, and he worked out the problem of the velocity of the nervous impulse both in the motor nerves of the frog and in the sensory nerves of man.

    0
    0
  • He studied the phenomena of electrical oscillations from 1869 to 1871, and in the latter year he announced that the velocity of the propagation of electromagnetic induction was about 314,000 metres per second.

    0
    0
  • Fitzgerald was the first to attempt to measure the length of electric waves; Helmholtz put the problem into the hands of his favourite pupil, Heinrich Hertz, and the latter finally gave an experimental demonstration of electromagnetic waves, the "Hertzian waves," on which wireless telegraphy depends, and the velocity of which is the same as that of light.

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  • To him is also due the discovery of the power of rotatory polarization exhibited by quartz, and last of all, among his many contributions to the support of the undulatory hypothesis, comes the experimentum crucis which he proposed to carry out for comparing directly the velocity of light in air and in water or glass.

    0
    0
  • On the emission theory the velocity should be accelerated by an increase of density in the medium; on the wave theory, it should be retarded.

    0
    0
  • In 1838 he communicated to the Academy the details of his apparatus, which utilized the revolving mirrors employed by Sir C. Wheatstone in 1835 for measuring the velocity of the electric discharge; but owing to the great care required in the carrying out of the project, and to the interruption to his labours caused by the revolution of 1848, it was the spring of 1850 before he was ready to put his idea to the test; and then his eyesight suddenly gave way.

    0
    0
  • They appealed also to the velocity and dilatation of aeriform bodies, to whirlwinds and inflated balloons.

    0
    0
  • He also calculated the effect of surface-tension on the propagation of waves on the surface of a liquid, and determined the minimum velocity of a wave, and the velocity of the wind when it is just sufficient to disturb the surface of still water.

    0
    0
  • The edge of the drop is drawn out by the surface-tension of A with a force greater than the sum of the tensions of the two surfaces of the drop. The drop, therefore, spreads itself out, with great velocity, over the surface of A till it covers an enormous area, and is reduced to such extreme tenuity that it is not probable that it retains the same properties of surface-tension which it has in a large mass.

    0
    0
  • If we place a small floating body in a shallow vessel of water and wet one side of it with alcohol or ether, it will move off with great velocity and skim about on the surface of the water, the part wet with alcohol being always the stern.

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  • When a liquid flows out of a vessel through a circular opening in the bottom of the vessel, the form of the stream is at first nearly cylindrical though its diameter gradually diminishes from the orifice downwards on account of the increasing velocity of the liquid.

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  • The resistance of the air produces little disturbance until the velocity becomes very great.

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  • In consequence of the transformation being in a more advanced stage at the forward than at the hinder end, the ligament remains for a moment connected with the mass behind, when it has freed itself from the mass in front, and thus the resulting spherule acquires a backwards relative velocity, which of necessity leads to a collision.

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  • Under the action of a vibrator of suitable periodic time the resolution is regularized, and then each drop, breaking away under like conditions, is projected with the same velocity, and therefore follows the same path.

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  • In the case of a continuous jet, the equation of continuity shows that as the jet loses velocity in ascending, it must increase in section.

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  • Now it is shown in hydrodynamics that she velocity of propagation of waves in deep water is that acquired by a heavy body falling through half the radius of the circle whose circumference is the wave-length, or _ f_X _ ga 27rT 'I ' v2- 2r 2r pn This velocity is a minimum when X=2.7r gp' and the minimum value is v= 4 - p g For waves whose length from crest to crest is greater than X, the principal force concerned in the motion is that of gravitation.

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  • When a small body is partly immersed in a liquid originally at rest, and moves horizontally with constant velocity V, waves are propagated through the liquid with various velocities according to their respective wave-lengths.

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  • In front of the body the relative velocity of the fluid and the body varies from V where the fluid is at rest, to zero at the cutwater on the front surface of the body.

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  • The waves produced by the body will travel forwards faster than the body till they reach a distance from it at which the relative velocity of the body and the fluid is equal to the velocity of propagation corresponding to the wave-length.

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  • Then comes the stationary wave of minimum velocity, which is the most marked of the series.

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  • If the current-function of the water referred to the body considered as origin is Ili, then the equation of the form of the crest of a wave of velocity w, the crest of which travels along with the body, is d =w ds where ds is an element of the length of the crest.

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  • To integrate this equation for a solid of given form is probably difficult, but it is easy to see that at some distance on either side of the body, where the liquid is sensibly at rest, the crest of the wave will approximate to an asymptote inclined to the path of the body at an angle whose sine is w/V, where w is the velocity of the wave and V is that of the body.

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  • But those whose wave-length is near to that of the wave of minimum velocity will diverge less than any of the others, so that the most marked feature at a distance from the body will be the two long lines of ripples of minimum velocity.

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  • If the angle between these is 20, the velocity of the body is w sec 0, where w for water is about 23 centimetres per second.

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  • In so far as the vibrations may be regarded as isochronous, the distance between consecutive corresponding points of the recurrent figure, or, as it may be called, the wavelength of the figure, is directly proportional to the velocity of the jet, i.e.

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  • The larger (and heavier), falling with greater velocity, overtook and collided with the smaller (and lighter), which were thereby forced upwards.

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  • The great velocity with which the wing is driven converts the impression or blur made by it into what is equivalent to a solid for the time being, in the same way that the spokes of a wheel in violent motion, as is well understood, more or less completely substance of the wing.

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  • The different parts of the wing, moreover, travel at different degrees of velocity - the tip and posterior margin of the wing always rushing through a much greater space, in a given time, than the root and anterior margin.

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  • This comes of the action and reaction of matter, the resistance experienced varying according to the density of the atmosphere and the shape, extent and velocity of the body acting upon it.

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  • To meet these peculiarities the insect, bird and bat are furnished with extensive flying surfaces in the shape of wings, which they apply with singular velocity and power to the air, as levers of the third order.

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  • While, therefore, there is apparently no correspondence between the area of the wing and the animal to be raised, there is, except in the case of sailing insects, birds and bats, an unvarying relation as to the weight and number of oscillations; so that the problem of flight would seem to resolve itself into one of weight, power, velocity and small surfaces, versus buoyancy, debility, diminished speed and extensive surfaces - weight in either case being a sine qua non.

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  • When such a model is wound up and let go it descends about 2 ft., after which, having acquired initial velocity, it rises and flies in a forward direction at a height of from 8 to io ft.

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  • It elevated itself from the ground with ease, and flew in a horizontal direction for a distance of 24 ft., and at a velocity of 20 m.

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  • The machine, fully prepared for flight, was started from the top of an inclined plane, in descending which it attained a velocity necessary to sustain it in its further progress.

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  • That velocity would be gradually destroyed by the resistance of the air to the forward flight; it was, therefore, the office of the steamengine and the vanes it actuated simply to repair the loss of velocity; it was made, therefore, only of the power a,nd weight necessary for that small effect."

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  • The idea was to get up the initial velocity by a preliminary run on the ground.

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  • In April the gradient is so slight that any temporary fall of pressure to the south of England or any temporary rise of pressure to the north, which would suffice in other months merely to reduce the velocity of the south-westerly wind, is sufficient in that month to reverse the gradient and produce an east wind over the whole country.

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  • The wind, however, rarely attains any exceptional velocity.

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  • At Turin it has an average width of 400 to 415 ft., a mean depth of 32 to 51 ft., and a velocity of i to 3 ft.

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  • Originally conquered by the fluvial deposits from the sea, it now stretches out as a vast dead level, in which the rivers find their velocity checked, and their current no longer able to carry along the silt which they have brought down from northern India.

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  • Bridges, more brilliant than the rest of the photosphere, form across them, and they may divide into two parts which separate from one another with great velocity.

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  • But this theory gives no clue to the results relating to hydrogen, which belongs to a high level, and which Adams has shown to move with an angular velocity decidedly greater than the equatorial angular velocity below it, and not to show any sign of falling off towards the poles.

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  • Lines of lanthanum and carbon which are believed to belong to a low level showed systematically smaller angular velocity than the average.

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  • The eruptive prominences, called also metallic, because it is they which show at their bases a complete bright line spectrum of the metallic elements, rush upwards at speeds which it is difficult to associate with transfers of matter; the velocity often exceeds loo m.

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  • This is remarkable only in point of velocity.

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  • Larmor suggests is due to relaxation of the spring of the surrounding ether by reason of the crowding of the molecules; a shift of 0.17 tenth-metres would, if interpreted by Doppler's principle, have been read as a receding velocity of I I km.

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  • Not so drop C, for directly the summit is passed the wind necessarily widens out vertically and, having a greater space to fill, loses forward velocity.

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  • The well itself must be lined; and its yield is therefore confined to such water as can be drawn through the sides or the bottom of the lining without setting up a sufficient velocity to cause any sand to flow with the water.

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  • The rate of increase of velocity towards any isolated aperture through which water passes into the side of a well sunk in a deep bed of sand is, in the neighbourhood of that aperture, inversely proportional to the square of the distance therefrom.

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  • Thus, the velocity across a little hemisphere of sand only z in.

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  • This process, in effect, leaves each orifice surrounded by a hemisphere of coarse sand across which the water flows with comparative freedom from a larger hemisphere where the corresponding velocity is very slow, and where the presence of finer and more obstructive particles is therefore unimportant.

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  • From the overflow sill the bye-wash channel may be gradually narrowed as the crest of the embankment is passed, the water being prevented from attaining undue velocity by steps of heavy masonry, or, where the gradient is not very steep, by irregularly set masonry.

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  • The rim is also subject to a centrifugal tension of amount wv 2 /g pounds per square inch of section, where w is the weight in pounds of a length of one foot of the pulley rim one square inch in section, and v is the velocity of the rim in feet per second.

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  • This stress amounts to 1043 lb per square inch, if the velocity is loo ft.

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  • The combination of these stresses generally limits the rim velocity of cast-iron pulleys to 80 or loo ft.

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  • The magnitude of the unbalanced force, for a mass of w pounds at a radius of r feet and a velocity of v feet per second, is expressed by wv 2 /gr lb; and, since the force varies as the square of the velocity, it is necessary carefully to balance a pulley running at a high speed to prevent injurious vibrations.

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  • Other forms, which do not require so lengthy a chain, sometimes employ an epicyclic train to obtain the reduced velocity of the load.

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  • The flood season begins in March and continues till September, the average depth of the river rising from 9 to 24 ft., and the velocity of the current increasing from 3 to 7 m.

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  • If the velocity is sufficiently high the coal may be carried the whole length of a 20-ft.

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  • If M is the central mass, n the angular velocity, and a the distance, the balance of the two forces is expressed by the equation an' =.1111a2, whence a 3 n 2 = M, a constant.

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  • Thirdly, the entire vis viva of the system or, as it is now called, the energy, which is obtained by multiplying the mass of each body into half the square of its velocity, is equal to the sum of the quotients formed by dividing the product of every pair of the masses, taken two and two, by their distance apart, with the addition of a constant depending on the original conditions of the system.

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  • Another corollary is that in the case of a body moving in a parabolic orbit the velocity at any moment is that which would be acquired by the body in falling from an infinite distance to the place it occupies at the moment.

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  • If projected with this velocity in any direction the point of projection will be at the end of the minor axis of the orbit, because this is the only point of an ellipse of which the distance from the focus is equal to the semi-major axis of the curve, and therefore the only point at which the distance of the body from the sun is equal to its mean distance.

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  • It is an elementary principle of mechanics that this force varies directly as the product of the distance of the moving body from the centre of motion into the square of its angular velocity.

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  • To fix the ideas let us suppose that the additional attraction is only an impulse received at the moment of passing the point P. The first effect will evidently be to change either the velocity or the direction in which the planet is moving at the moment, or both.

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  • If, with the changed velocity we again compute the elements they will be different from the former elements.

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  • The position and velocity being given in all three co-ordinates, a certain osculating plane is determined for each instant in which the planet is moving at that instant.

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  • He further elaborated it by the introduction of " eccentrics," which accounted for the changes in orbital velocity of the sun and moon by a displacement of the earth, to a corresponding extent, from the centre of the circles they were assumed to describe.

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  • Pursuing the inquiry, he found that its velocity was uniform with respect to no single point within the orbit, but that the areas described, in equal times, by a line drawn from the sun to the planet were strictly equal.

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  • Again, since the constant of aberration defines the ratio between the velocity of light and the earth's orbital speed, the span of the terrestrial circuit, in other words, the distance of the sun, is immediately deducible from known values of the first two quantities.

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  • The pitch of a steam-whistle quite obviously rises and falls as the engine to which it is attached approaches and recedes from a stationary auditor; and light pulses are modified like sound-waves by velocity in the line of sight.

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  • Integrated light, accordingly, tells nothing about velocity; but analysed light does, when it includes bright or dark rays the normal positions of which are known.

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  • Not only the grosser facts concerning radial velocity, but variations in it so small as a mile, or less, per second, have been recorded and interpreted in terms of deep meaning.

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  • His theorem that a fluid issues from a small orifice with the same velocity (friction and atmospheric resistance being neglected) which it would have acquired in falling through the depth from its surface is of fundamental importance in hydraulics.

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  • It was designed of sufficient width, as was supposed, for the simultaneous passage of boats in opposite directions; but on account of the great velocity of the current this has been found to be impracticable.

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  • As the presentations yield to the pressure, the pressure itself diminishes, so that the velocity of sinking decreases, i.e.

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  • From Ansongo to Say, some 250 m., the river flows through several rocky passes, the current attaining great velocity.

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  • He perceived the analogy between the power which holds the moon in the neighbourhood of the earth, and compels Jupiter's satellites to circulate round their primary, and the attraction exercised by the earth on bodies at its surface; 1 but he failed to conceive the combination of central force with tangential velocity, and was disposed to connect the revolutions of the planets with the axial rotation of the sun.

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  • More valid instances of the anticipation of modern discoveries may be found in his prevision that a small annual parallax would eventually be found for some of the fixed stars, and that extra-Saturnian planets would at some future time be ascertained to exist, and in his conviction that light travels with a measurable, although, in relation to terrestrial distances, infinite velocity.

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  • The first law of motion - that which expresses the principle of inertia - is virtually contained in the idea of uniformly accelerated velocity.

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  • As the planet revolves around the centre, each radius vector describes a surface of which the area swept over in a unit of time measures the areal velocity of the planet.

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  • The constancy of this velocity in the case of the sun and a single planet is formulated in Kepler's second law.

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  • We shall thus have a projected areal velocity, the product of which by the mass of the planet is the moment of momentum of the latter.

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  • The average velocity of winds for the entire state for it years preceding 1906 was 9.8 m.

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  • Both at very high and very low pressures the coefficient of friction is affected by the intensity of pressure, and, just as with velocity, it can only be regarded as independent of the intensity and proportional simply to the total load within more or less definite limits.

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  • These experiments distinctly point to the conclusion, although without absolutely proving it, that in such cases the coefficient of kinetic friction gradually increases as the velocity becomes extremely small, and passes without discontinuity into that of static friction.

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  • According to the Cartesians, this quantity was directly proportional to velocity; according to their opponents, it varied with the square of the velocity.

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  • In this brief tract, Kant, apparently in entire ignorance of the explanation given in 1735 by Hadley, points out how the varying velocity of rotation of the successive zones of the earth's surface furnishes a key to the phenomena of periodic winds.

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  • The average velocity of the winds is comparatively low and violent storms are rare.

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  • As an example, consider the equation for the velocity v of an object that undergoes an acceleration a for a time t.

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  • Mean velocity measurements were taken by using a precision constant temperature linearized hot-wire anemometer designed for low speed water flows.

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  • The effect of throat just aft of the lip is derived from theoretical calculations taking account of the velocity profile.

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  • The rotation curve is a plot of the orbital velocity of the clouds around the galactic center vs. their distance from the Galaxy center.

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