If a point be in motion in any orbit and with any velocity, and if, at each instant, a line be drawn from a fixed point parallel and equal to the velocity of the moving point at that instant, the extremities of these lines will lie on a curve called the hodograph.
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.
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.
Every orbit must clearly have a hodograph, and, conversely, every hodograph a corresponding orbit; and, theoretically speaking, it is possible to deduce the one from the other, having given the other circumstances of the motion.
For applications of the hodograph to the solution of kinematical problems see Mechanics.
(See MECHANICS and HODOGRAPH.)
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.
In the motion of a projectile under gravity the hodograph is a vertical line described with constant velocity.
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.
Hence the hodograph is similar and similarly situated to the locus of Z (the Flo.
70 exhibits the various cases, with the hodograph in its proper orientation.
The pole 0 of the hodograph is inside on or outside the circle, according as the orbit is an ellipse, parabola or hyperbola.
In any case of a central orbit the hodograph (when turned through a right angle) is similar and similarly situated to the reciprocal polar of the orbit with respect to the centre of force.
Thus for a circular orbit with the centre of force at an excentric point, the hodograph is a conic with the pole as focus.
There is also the extremely ingenious invention of the hodograph.
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).
that at any point the tangent to the hodograph is parallel to the direction, and the velocity in the hodograph equal to the magnitude of the resultant acceleration at the corresponding point of the orbit.
Phil.): - Let x, y, z be the coordinates of P in the orbit,, r t, those of the corresponding point T in the hodograph, then dx dy _ dz c= ' 71 - a' - at therefore Also, if s be the arc of the hodograph, ds = v = V V1 1) j dt + (dt2) dt Equation (1) shows that the tangent to the hodograph is parallel to the line of resultant acceleration, and (2) that the velocity in the hodograph is equal to the acceleration.
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