Velocities Sentence Examples

Thus, the ratio of the losses at the two ends is two to one - the same as the ratio of the assumed ionic velocities.

The velocities in air and water being respectively 1090 and 4700 ft.

The result has been that wind velocities published in many official publications have of ten been in error by nearly 5 0%.

He initiated in 1866 the spectroscopic observation of sunspots; applied Doppler's principle in 1869 to determine the radial velocities of the chromospheric gases; and successfully investigated the chemistry of the sun from 1872 onward.

There is reason to believe that in certain cases such complex ions do exist, and interfere with the results of the differing ionic velocities.

On the theory that the phenomena are wholly due to unequal ionic velocities this result would mean that the cation like the anion moved against the conventional direction of the current.

The velocity with which the ions move past each other is equal to the sum of their individual velocities, which can therefore be calculated.

Now Hittorf's transport number, in the case of simple salts in moderately dilute solution, gives us the ratio between the two ionic velocities.

Hence the absolute velocities of the two ions can be determined, and we can calculate the actual speed with which a certain ion moves through a given liquid under the action of a given potential gradient or electromotive force.

The details of the calculation are given in the article Electric conduction, § where also will be found an account of the methods which have been used to measure the velocities of many ions by direct visual observation.

The results go to show that, where the existence of complex ions is not indicated by varying transport numbers, the observed velocities agree with those calculated on Kohlrausch's theory.

His treatise was remarkable, not only as offering a satisfactory explanation of the coincidence between the lunar periods of rotation and revolution, but as containing the first employment of his radical formula of mechanics, obtained by combining with the principle of d'Alembert that of virtual velocities.

This will be evident if we consider that, since radii vectores of the hodograph represent velocities in the orbit, the elementary arc between two consecutive radii vectores of the hodograph represents the velocity which must be compounded with the velocity of the moving point at the beginning of any short interval of time to get the velocity at the end of that interval, that is to say, represents the change of velocity for that interval.

But as he was unacquainted with the law of the velocities of running water as depending upon the depth of the orifice, the want of precision which appears in his results is not surprising.

At a time when the Cartesian system of vortices universally prevailed, he found it necessary to investigate that hypothesis, and in the course of his investigations he showed that the velocity of any stratum of the vortex is an arithmetical mean between the velocities of the strata which enclose it; and from this it evidently follows that the velocity of a filament of water moving in a pipe is an arithmetical mean between the velocities of the filaments which surround it.

He supposed that the surface of the fluid, contained in a vessel which is emptying itself by an orifice, remains always horizontal; and, if the fluid mass is conceived to be divided into an infinite number of horizontal strata of the same bulk, that these strata remain contiguous to each other, and that all their points descend vertically, with velocities inversely proportional to their breadth, or to the horizontal sections of the reservoir.

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.

The real velocities derived from good observations are rarely, if ever, under 7 or 8 m.

In 1738 appeared his Hydrodynamica, in which the equilibrium, the pressure, the reaction and varied velocities of fluids are considered both theoretically and practically.

Reynolds, in his investigation, introducing no new form of law of distribution of velocities, uses a linear quantity, proportional to the mean free path of the gaseous molecules, which he takes to represent (somewhat roughly) the average distance from which molecules directly affect, by their convection, the state of the medium; the gas not being uniform on account of the gradient of temperature, the change going on at each point is calculated from the elements contributed by the parts at this particular distance in all directions.

If each of the fractions (3) is put equal to i/4h, it is readily found, from the first property of the normal state, that, of the s molecules of the first kind, a number sal (h3m3 /13)e hm (u2+v2+w2)dudvdw (4) Velocities.

T =273, we obtain R =1.35 X Io -16 and this enables us to determine the mean velocities produced by heat motion in molecules of any given mass.

In 1868 he proved incandescent carbon-vapours to be the main source of cometary light; and on the 23rd of April in the same year applied Doppler's principle to the detection and measurement of stellar velocities in the line of sight.

Comparing the velocities of sound U i and U2 in two different gases with densities and at the same temperature and pressure, and with ratios of specific heats 'yl, 72, theory gives Ui/U2 = 1/ {71 p 2/72 p i }.

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.

If the velocities of source and receiver are equal then the frequency is not affected by their motion or by the wind.

But if their velocities are different, the frequency of the waves received is affected both by these velocities and by that of the wind.

The velocities in different gases may be compared by this apparatus by filling the dust-tube with the gases in place of air.

To form stationary waves two equal trains must be able to travel in opposite directions with equal velocities, and to be superposed.

But this theory is very far from being of practical value for most purposes of gunnery; so that a first requirement is an accurate experimental knowledge of the resistance of the air to the projectiles employed, at all velocities useful in artillery.

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)].

The latter work contains elaborate investigations in regard to the centre of gravity, and it is remarkable also for the employment of the principle of virtual velocities.

But the main purport of the treatise was the exposition of an elaborate system of celestial harmonies depending on the various and varying velocities of the several planets, of which the sentient soul animating the sun was the solitary auditor.

Robins also made a number of important experiments on the resistance of the air to the motion of projectiles, and on the force of gunpowder, with computation of the velocities thereby communicated to projectiles.

The mechanical idea, named the parallelogram of velocities, permits a ready and easy graphical representation of these facts.

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.

The velocities ranged from about 400 to 1900 metres, the metals of small atomic weight giving as a rule the higher velocities.

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.

While some of the phenomena seem to indicate that the projection of metallic vapours into the centre of the spark is a process of molecular diffusion independent of the mechanism of the discharge, the different velocities obtained with bismuth, and the probability that the vibrating systems are not electrically neutral, seem to indicate that the projected metallic particles are electrified and play some part in the discharge.

Stark discovered that in the case of the series spectrum of hydrogen and of other similar spectra the lines were displaced indicating high velocities; in other cases no displacements could be observed.

The importance of a study of the changes of the vis viva depending on squares of velocities, or what is now called the "kinetic energy" of a system, was recognized in Newton's time, especially by Leibnitz; and it was perceived (at any rate for special cases) that an increase in this quantity in the course of any motion of the system was otherwise expressible by what we now call the "work" done by the forces.

Now the velocities u and v of the opposite ions under unit potential gradient, and therefore U and V under unit force, are known from electrical data.

On the kinetic theory the molecules of a gas are relatively far apart and there is nothing analogous to friction between two adjacent layers A and B moving with different velocities.

This action and reaction between layers in relative motion is equivalent to a frictional stress tending to equalize the velocities of adjacent layers.

The intervals to which the velocities referred were usually from five to ten minutes, but varied widely.

Spectroscopic examination had already suggested prodigious velocities of the order of woo m.

The above are velocities transverse to the line of sight.

The greatest radial velocities that have yet been found are about 60 m.

It seems difficult to account for the very remarkable and unsymmetrical distribution of the motions, unless we suppose that the stars form two more or less separate systems superposed; and it has been found possible by assuming two drifts with suitably assigned velocities to account very satisfactorily for the observed motions.

The velocities of the drifts differ considerably, the one whose apex is in Ophiuchus having about 1 2 times the speed of the other.

Until the hypothesis has been thoroughly tested by an examination of the line-of-sight velocities of stars from the same point of view, this physical interpretation must be received with some degree of caution; but there can be no doubt of the reality of the anomalies in the statistical distribution of proper motions of the stars, and of these it offers a simple and adequate explanation.

The step he took is really nothing more than the kinematical principle of the composition of linear velocities, but expressed in terms of the algebraic imaginary.

This is (in part) the celebrated principle of virtual velocities, now often described as the principle of virtual work, enunciated by John Bernoulli (1667-1748).

The velocities referred to are the velocities of the various points of the body in any imagined motion of the body through the position in question; they obviously bear to one another the same ratios as the corresponding infinitesimal displacements.

The instantaneous centre of CD will be at the intersection of AD, BC, and if CD be drawn parallel to CD, the lines CC, DD may be taken to represent the virtual velocities of C, D turned each through B a right angle.

A curve with I as abscissa and u as ordinate is called the curve of velocities or velocity-time curve.

The curve of positions corresponding to (4) is a parabola, and that of velocities is a straight line.

If two masses m1, nil moving in the same straight line impinge, with the result that the velocities are changed from u1, u2, to ui, ui, then, since the impulses on the two bodies must be equal and opposite, the total momentum is unchanged, i.e.

Obviously OV is parallel to the tangent to the path atP, and its magnitude is ds/dt, where s is the arc. If we project OV on the co-ordinate axes (rectangular or oblique) in the usual manner, the projections u, v, w are called the component velocities parallel to the axes.

If it, v be the component velocities at P along and perpendicular to OP (in the direction of 0 increasipg), FIG.

Again, the velocities parallel and perpendicular to OP change in the time & from it, v to uvb0, v+ubG, ultimately.

Again, if x, y, z be the co-ordinates of P, the component velocities of m are qzry, rxpz, pyqx, (6)

The motion of a rigid body in the most general case may be specified by means of the component velocities u, v, w of any point 0 of it which is taken as base, and the component angular velocities p, q, r.

The component velocities of any point whose co-ordinates relative to 0 are x, y, z are then u+qzry, v+rxpz, w+Pyqx (12)

If p, q, r be the component angular velocities about the principal axes at 0, we have (Ap+B2q+C,2)/r = (Ap+Bq1+Cr2)/2T, (3) each side being in fact equal to unity.

The moving axes Ox, Oy, 01 form a rigid frame of reference whose motion at time t may be specified by the three component angular velocities p, q, r.

However slight these forces may be, the total energy T+V must continually diminish so long as the velocities qi,q2,.

In the case of multiple freedom, the equations of small motion when modified by the introduction of terms proportional to the velocities are of the type daT -.

The terms due to F in (33) are such as would arise from frictional resistances proportional to the absolute velocities of the particles, or to mutual forces of resistance proportional to the relative velocities; they are therefore classed as frictional or dissipative forces.

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.

The coupling rod remains always parallel to itself, and all its points describe equal and similar circles relatively to the frame of the engine, and move in parallel directions with equal velocities at the same instant.

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.

Hence also the ratio of the com ponents of the velocities of two points A and B in the directions AP and BW respectively, both in the plane of rotation, is equal to the ratio of the perpendiculars Fni and Fn.

It is required to find the ratios of those angular velocities.

General Principle.From the definition of a line of conneyfion it follows that the components of the velocities of a pair of connected points along their line of connexson are equal.

And from this, and from the property of a rigid body, already stated in 29, it follows, that the components along a is ne of connection of all the points traversed by that line, whether -in the driver or in the follower, are equal; and consequently, that the velocities of any pair of points traversed by a line of connection are to each other inversely as the cosines, or directly as the secants, of the angles made by the paths of those points with the line of connection.

Also TA represents the equal components of the velocities of the FIG.

Application to a Pair of TurnIng Fseces.Let ai, a2 be the angular velocities of a pair of turning pieces; Of, Oi the angles which their line of connection makes with their respective planes of rotation; Ti, r2 the common perpendiculars let fall from the line of connection upon the respective axes of rotation of the pieces.

In the case of two shifting pieces this would involve equal and parallel velocities for all the points of each piece, so that there could be no rolling, and, in fact, the two pieces would move like one; hence, in the case of roIling contact, either one or both of the pieces must rotate.

That the angular velocities of a pair of turning pieces in rolling contact must be inversely as the perpendicular distances of any pair of points of contact from the respective axes.

Hence, in any pair of circular wheels which work together, the numbers of teeth in a complete circumference are directly as the radii and inversely as the angular velocities.

The only modification required in the formulae is, that in equation (26) the difference of the angular velocities should be substituted for their sum.

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.

The angular velocities of the screws are inversely as their numbers of threads.

The angular velocities of a pair of connected circular pulleys or drums are inversely as the effective radii.

Coupling of Parallel Axes.Two or more parallel shafts (such as those of a locomotive engine, with two or more pairs of driving wheels) are made to rotate with constantly equal angular velocities by having equal cranks, which are maintained parallel by a coupling-rod of such a length that the line of c000exion is equal to the distance between the axes.

The line of intersection of the planes perpendicular to the paths of the two connected points at a given instant is the instantaneous axis of the link at that instant; and the velocities of the connected points are directly as their distances from that axis.

Let ai, a2, af be the angular velocities of the first, intermediate, and last shaft in this train of two Hookes couplings.

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.

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.

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.

It should be understood that the instantaneous centre considered in the preceding paragraphs is available only for estimating relative velocities; it cannot be used in a similar manner for questions regarding acceleration.

Principle of the Equality of Energy and Work.FroIn the first law of motion it follows that in a machine whose pieces move with uniform velocities the efforts and resistances must balance each other.

This principle, applied to a machine whose parts move with uniform velocities, is equivalent to saying that in any given interval of time the energy exerted is equal to the work performed.

The principles of this reduction are that the ratio of the given to the equivalent force is the reciprocal of the ratio of the velocities of their points of application, and the ratio of the given to the equivalent couple is the reciprocal of the ratio of the angular velocities of the pieces to which they are applied.

The principle which he adopted is that of virtual velocities, a principle which under his hands was gradually transforming itself into what is now known as the principle of the conservation of energy.

The analytical expression for the motion in the latter case involves exponential terms, one of which (except in case of a particular relation between the initial displacements and velocities) increases rapidly, being equally multiplied in equal times.

In this case drops which break away with different velocities are carried under the action of gravity into different paths; and thus under ordinary circumstances a jet is apparently resolved into a " sheaf," or bundle of jets all lying in one vertical plane.

If, for example, besides the principaldisturbance,whichdetermines the size of the drops, there be another of twice the period, it is clear that the alternate drops break away under different conditions and therefore with different velocities.

Such collisions are inevitable in consequence of the different velocities acquired by the drops under the action of the capillary force, as they break away irregularly from the continuous portion of the jet.

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.

But as the head increases, so do the lateral velocities which go to form the transverse vibrations.

The atmosphere, because of its great tenuity, mobility and comparative imponderability, presents little resistance to bodies passing through it at low velocities.

It is in fact an impossibility that loss should go on without regeneration, for if any part of the sun's body loses heat, it will be unable to support the pressure of neighbouring parts upon it; it will therefore be compressed, in a general sense towards the sun's centre, the velocities of its molecules will rise, and its temperature will again tend upwards.

It is clear that these results may give a simple key to some puzzling anomalies, and on the other hand, they may throw a measure of uncertainty over absolute determinations of line-of-sight velocities.

Attempts have been made, by adding certain coagulants to the water to be filtered, to increase the power of sand and other granular materials to arrest bacteria when passing through them at much higher velocities than are possible for successful filtration by means of the surface film upon sand.

In the language of algebra putting m l, m2, m 3, &c. for the masses of the bodies, r1.2 r1.3 r2.3, &c. for their mutual distances apart; vi, v 2, v 3, &c., for the velocities with which they are moving at any moment; these quantities will continually satisfy the equation orbit, appear as arbitrary constants, introduced by the process of integration.

Putting a, b, c, d, for the constants, the general form of the solution will be x = fl (a,b,c,d,t) y = f2(a,b,c,d,t) From these may be derived by differentiation as to t the velocities dt =f '1(a,b,c,d,t) = x'  ?

He employed in his discussion the radial velocities of 280 stars, spectroscopically determined; and the upshot signally exemplified the community of interests between the rising science of astrophysics and the ancient science of astrometry.

This we can only do by conceiving them as originally moving through intelligible space in rectilinear paths and with uniform velocities.

From the leaning tower of Pisa he afforded to all the professors and students of the university ocular demonstration of the falsehood of the Peripatetic dictum that heavy bodies fall with velocities proportional to their weights, and with unanswerable logic demolished all the time-honoured maxims of the schools regarding the motion of projectiles, and elemental weight or levity.

The ebb and flow of the tides were, he asserted, a visible proof of the terrestrial double movement, since they resulted from inequalities in the absolute velocities through space of the various parts of the earth's surface, due to its rotation.

Conceiving that the simplest principle is the most likely to be true, he assumed as a postulate that bodies falling freely towards the earth descend with a uniformly accelerated motion, and deduced thence that the velocities acquired are in the direct, and the spaces traversed in the duplicate ratio of the times, counted from the beginning of motion; finally, he proved, by observing the times of descent of bodies falling down inclined planes, that the postulated law was the true law.

Even here, he was obliged to take for granted that the velocities acquired in descending from the same height along planes of every inclination are equal; and it was not until shortly before his death that he found the mathematical demonstration of this not very obvious principle.

He gave the first satisfactory demonstration of equilibrium on an inclined plane, reducing it to the level by a sound and ingenious train of reasoning; while, by establishing the theory of "virtual velocities," he laid down the fundamental principle which, in the opinion of Lagrange, contains the general expression of the laws of equilibrium.

In his Discorso intorno alle cose the stanno su l'acqua, published in 1612, he used the principle of virtual velocities to demonstrate the more important theorems of hydrostatics, deducing from it the equilibrium of fluid in a siphon, and proved against the Aristotelians that the floating of solid bodies in a liquid depends not upon their form, but upon their specific gravities relative to such liquid.

For many practical purposes these statements are sufficiently accurate, and they do in fact sensibly represent the results of experiment for the pressures and at the velocities most commonly occurring.

To measure relative drift velocities of small and large floes due to wind and wave action.

By detecting the gamma rays produced, it is possible to measure the electron velocities in the sample under review.

The specially contoured housing further improves the meters linearity particularly at lower fluid velocities.

Thus, any triangle of velocities could not direct a small passerine into our longitude.

The slower propagation of light in gas or water than in air or vacuum may be attributed to a greater density, or to a less rigidity, in the former case; or we may adopt the more complicated supposition that both these quantities vary, subject only to the condition which restricts the ratio of velocities to equality with the known refractive index.

Simpson observed a fall in q for wind velocities exceeding 2 on Beaufort's scale.

Now it is probable that the main cause of oceanic circulation is the driving force of the winds upon the superficial layers of water; hence periodic and irregular changes in the direction and velocities of ocean currents are probably due to changes in atmospheric circulation traceable to changes in the quantities of heat absorbed from the sun by the earth's atmosphere.

This conclusion is confirmed by the results of the direct visual determination of ionic velocities (see Conduction, Electric, § Ii.), which, in cases where the transport number remains constant, agree with the values calculated from those numbers.

His removal of the considerable discrepancy between the actual and Newtonian velocities of sound,' by taking into account the increase of elasticity due to the heat of compression, would alone have sufficed to illustrate a lesser name.

Let PP1P2 be the path of the moving point, and let OT, OT 1, OT2, be drawn from the fixed point 0 parallel and equal to the velocities at P, P 1, respectively, then the locus of T is the hodograph of the orbits described by P (see figure).

Torricelli, observing that in a jet where the water rushed through a small ajutage it rose to nearly the same height with the reservoir from which it was supplied, imagined that it ought to move with the same velocity as if it had fallen through that height by the force of gravity, and hence he deduced the proposition that the velocities of liquids are as the square root of the head, apart from the resistance of the air and the friction of the orifice.

He assumed that the distribution of molecules and of their velocities, at each point, was slightly modified, from the exponential law belonging to a uniform condition, by the gradient of temperature in the gas (see Diffusion).

Moreover, Galileo recognized, to some extent at any rate, the principle of simple superposition of velocities and accelerations due to different sets of circumstances, when these are combined (see Mechanics).

But the fact that the apparent rapidity of motion of this phantom may exceed in any ratio that of the spectator is of importance - enabling us to see how velocities, apparently of impossible magnitude, may be accounted for by the mere running along of the condition of visibility among a group of objects no one of which is moving at an extravagant rate.

In order to obtain at all events a qualitative representation of these it is usual to introduce into thc equations frictional terms proportional to the velocities.

Comparative Motion of Two Pistons.If there be but two pistons, whose areas are af and af, and their velocities Vf and vI, their comparative motion is expressed by the equation V2/Vf = aia/2; (2)

Trains of Wheelwork.Let A1, A2, A3, &c., A,,,_1, A,,, denote a series of axes, and aj, a1, a3, &c., a,,,1, a,,, their angular velocities.

But there are cases in which the motions of both bodies are appreciable, and must be taken into accountsuch as the projection of projectiles, where the velocity of the recoil or backward motion of the gun bears an appreciable proportion to the forward motion of the projectile; and such as the propulsion of vessels, where the velocity of the water thrown backward by the paddle, screw or other propeller bears a very considerable proportion to the velocity of the water moved forwards and sideways by the ship. In cases of this kind the energy exerted by the effort is distributed between the two bodies between which the effort is exerted in shares proportional to the velocities of the two bodies during the action of the effort; and those velocities are to each other directly as the portions of the effort unbalanced by resistance on the respective bodies, and inversely as the weights of the bodies.

He also discovered the remarkable fact that the parabolas described (in a vacuum) by indefinitely numerous projectiles discharged from the same point with equal velocities, but in all directions have a paraboloid of revolution for their envelope.

He found that dilution with water does not effect proportionate alteration in the transpiration velocities of different liquids, and a certain determinable degree of dilution retards the transpiration velocity.

To find Hubble 's constant we have to apply Hubble 's law to objects whose distances and recession velocities are already known.

A pinch of the proper powder pushes this bullet out the muzzle at subsonic velocities, thus permitting silent fire.

The precise location of the initial stage of development of the Ness Pole must be linked to where tidal current velocities start to diminish.

In the example of Figure 3, tho, the sedimentary P-wave velocities are significantly lower than the basalt S-wave velocities.

Most meridional propagation velocities from high to low latitudes are less than 600 m/s.

The method originally used by Huggins, who first conceived and proved the possibility of measuring stellar velocities in the line of sight, was to measure with a filar micrometer the displacement of some well-known line in a stellar spectrum relative to the corresponding line of a terrestrial spectrum.

In the experiment imagined by Lord Rayleigh a porous diaphragm takes the place of the partition and trap-doors imagined by Clerk Maxwell, and the molecules sort themselves automatically on account of the difference in their average velocities for the two gases.

In such experiments the molecular energy of a gas is converted into work only in virtue of the molecules being separated into classes in which their velocities are different, and these classes then allowed to act upon one another through the intervention of a suitable heat-engine.

These atoms, differing only in size, figure and weight, are perpetually moving with equal velocities, but at a rate far surpassing our conceptions; as they move, they are for ever giving rise to new worlds; and these worlds are perpetually tending towards dissolution, and towards a fresh series of creations.