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Acceleration of motion of the second at any instant; and, there being no fixed centre. velocity perpendicular to this plane at starting, there is there
fore none throughout the motion; thus the point moves in the
at - y de
- 0, etc.,
= C,, dt
-Y dt dt
dt dt Hence at once C,x+C2y + z = 0, or the motion is in a plane through the origin. Take this as the plane of xy, then we have only the one equation
=C,= h (suppose).
- y dt
if A be the area intercepted by the curve, a fixed radius-vector, and the radius-vector of the moving point. Hence the area in
creases uniformly with the time. b. In the same case the velocity at any point is inversely as the perpendicular from the fixed point upon the tangent to the path, the momentary direction of motion.
For evidently the product of this perpendicular and the velocity gives double the area described in one second about the fixed point.
Or thus—if p be the perpendicular on the tangent,
dy dx P= x
ds dy dx and therefore
Acceleration directed to a fixed centre.
If we refer the motion to co-ordinates in its own plane, we have ouly the equations
dx Px d'y_Py
do whence, as before,
dt If, by the help of this last equation, we eliminate t from dx Pac
substituting polar for rectangular co-ordinates, we de arrive at the polar differential equation of the path. For variety, we may derive it from the formula of $ 32.
dor do\ They give
dt 1 Putting
(a) * = P,
= U, we have
1/dA Also = h'u, the substitution of which values gives us
Р d69 hu
(1), the equation required. The integral of this equation involves two arbitrary constants besides h, and the remaining constant belonging to the two differential equations of the second order above is to be introduced on the farther integration of
Other examples of these principles will be met with in the
chapters on Kinetics. Hodograph. 37. If from any fixed point, lines be drawn at every instant,
representing in magnitude and direction the velocity of a point describing any path in any manner, the extremities of these lines form a curve which is called the Hodograph. The invention of this construction is due to Sir W. R. Hamilton. One of the most beautiful of the many remarkable theorems to which it led him is that of g 38.
Since the radius-vector of the hodograph represents the velocity at each instant, it is evident ($ 27) that an elementary arc represents the velocity which must be compounded with the velocity at the beginning of the corresponding interval of time, to find the velocity at its end. Hence the velocity in the hodograph is equal to the acceleration in the patb; and the tangent to the hodograph is parallel to the direction of the acceleration in the path.
If x, y, z be the co-ordinates of the moving point, $, *, & those of the corresponding point of the hodograph, then evidently
de dn dg and therefore
dto dt dta
C)* + C)
dt dt dt2 or the velocity in the hodograph is equal to the rate of accelera
tion in the path. Hodograph
38. The hodograph for the motion of a planet or comet is comet, deo" always a circle, whatever be the form and dimensions of the orbit. Kepler's In the motion of a planet or comet, the acceleration is directed laws.
towards the sun's centre. Hence ($ 36, 6) the velocity is in
versely as the perpendicular from that point upon the tangent Hodograph
For a projectile unresisted by the air, it will be shewn in
dt de -9,
For the case of a planet or comet, instead of assuming as Hodograph
dx px d'y _ My
duced from Newton's law of force.
* See our smaller work, $ 51.
Hence for the hodograph
the three first dt
among integrals (1), (2), (3) above. We thus get
-h+ Ay - Bx =
tions of the
a conic section of which the origin is a focus.
increasing distance according to the same law as gravitation.
40. If two points move, each with a definite uniform velo-
Curves of pursuit.