A

B

R the impulse between them, measured by the momentum generated (Art. 151), which must be the same for each, and in opposite directions, by the Third Law of Motion (Art. 110); so that if A impinges on B, the impulse R increases B's motion and diminishes A's. As the balls are inelastic, they do not separate after the impact, therefore they move with a common velocity; let this be v.

R

R

Before the impact the momenta of A and B are respectively MV, M'V';

therefore after the impact, the momentum of A = MV – R, and that of BM'V' + R ;

156. From the equation Mv+M'v = MV + M'V' we see that the momentum of the system before impact is equal to the momentum afterwards.

157. If the balls be elastic, we have to consider (1) the circumstances of their mutual compression, (2) the circumstances of their restitution of figure. We have then two impulses,

L.

N

R for the compression, and R' for the restitution of figure, the whole impulse being R+R', which increases B's motion and diminishes A's. During the compression the impulse R' has not been acting, and therefore the circumstances are the same as if the balls were inelastic; therefore R is the same in this case as MM' in the former, and consequently=

M+M' (V−V').

Let v, v' be the velocities of A and B after impact; therefore their momenta after impact are Mv, M'v';

It is also supposed that R' bears to R a ratio depending only upon the nature of the substances that impinge, so that R'=eR, where e depends only upon the nature of the materials of which A and B are composed. e is called the modulus of elasticity, and lies between 0 and 1: if e 1, the elasticity is said to be perfect, but we know of no such case in nature.

or the velocity of separation of the balls after impact the velocity of approach before impact :: e: 1.

=

This is a property that can be tested by observation, and establishes the correctness of our hypothesis that R' eR. For if we had started with the equations

and assumed that v'-v=e(V— V'), we should directly have obtained R' = eR.

159. We have also Mv+ M'v' = MV+M'V' in this case as well as in the preceding, so that no momentum is lost by the impact.

160. If x, x' be the distances of the centers of the balls from any fixed point in the line of impact at time t after the collision, we have

And if a be the distance of the center of gravity of the two from the same point,

therefore the velocity of the center of gravity of the two balls

after impact is

MV+M'V'

M+M''

the same as its velocity before

impact.

161. If the impact be not direct, i. e. if the balls be initially moving in directions not coincident with the line of impulse, we must resolve their velocities into the directions of the line of impulse and perpendicular to it. The resolved parts in the direction of impulse will be affected by the impact after the manner of the preceding investigations: those perpendicular to that direction will not be affected at all, because the balls are smooth. Then the velocities of either ball in the direction perpendicular to the line of impulse before and after the impact are equal; and the equations of the preceding Articles will be true in this case, wherein V, V', v, v', represent the resolved parts of the velocities in the direction of impact.

162. In the collision of two perfectly elastic balls, the vis viva of the system after impact is the same as that before impact.

By the vis viva of the system is meant the sum of the vires viva of the balls. (See Art. 109.)

We have, using the notation of Art. 157,

because the elasticity is perfect;

and Mv+M'v' = MV + M' V'.

Whence M(v-V) = M' (V' — v'),

and v+ VV' + v';

.. M(v2 — V2) = M' (V12 — v'2),

or My2+M'v" = MV2+ M'V'.

163. If the elasticity be imperfect, vis viva is lost by the collision.

·._M (v2 — V2) = M' ( V12 — v'2) + (1 − e) M' ( V' — v′) (V — 'V'′).

(V− V');

(V-V'),

.. M (v2 — V2) = M' (V12 — v'2) − (1 − e) M'.

which is less than MV + M'V", e<1.

164. An elastic ball strikes a smooth plane obliquely: to determine the motion.

If a plane be drawn through the initial direction of motion and the normal to the plane against which the ball impinges, there is no velocity and no impulse perpendicular to this plane, and therefore after impact the ball will still move in it. Let this plane be that in which the annexed figure is drawn, so that the plane against which the ball impinges is perpendicular to that of the paper.

Let v, v' be the velocities of the ball before and after impact,

M its mass,

(1+e) R the impulse,

Ө

0, 4, the angles which the directions of motion before and after impact make with the normal to the plane.

Then as the impulse does not affect the velocity resolved in the direction of the plane, we have

v sin = v' sin p.

For the motion perpendicular to the plane, we have