If the balls before impact are moving in opposite directions, we have merely to give opposite signs to Vand v.

It remains to determine the impulsive pressure between the two balls. We have determined the velocity of each after impact on the assumption that the changes of their respective momenta are equal and opposite. To find the impulse we need only consider the change of momentum of one of the balls.

The velocity of the first before impact is V, and after impact its velocity is V1, hence the change of its momentum is M (V- V1), that is

Hence if I denote the impulsive pressure between the balls, we have

We may notice that if the masses of the balls are equal and the elasticity perfect, or e equal to 1, it follows from equations (IV) and (V) that v1 = V and V=v, or the balls exchange their velocities. If in the equations we make m infinite we obtain the same result as in Art. 123, in fact we revert to the case of impact against a moving obstacle whose velocity is unchanged by the collision.

127. We proceed now to consider the case of the impact of two smooth spheres not moving in the same or in opposite directions. We shall investigate only the case in which the centres of the two spheres are moving in the same plane. The solution of the general case is precisely similar, but the geometry is more difficult.

Let the two spheres be called A and B respectively, M denoting the mass of A, and m that of B.

Let KG be the straight line drawn through O, O' the centres of the spheres at the moment of impact. Let DO be

1

the direction in which the sphere A is moving before impact, and its velocity, EO' the direction of motion of B before impact, and v its velocity. Let V1, v, be their respective velocities after impact, and OH, O'L the directions in which they are respectively moving. Let the angles DOK, EOK, HOĞ, LOG be denoted by a, B, 0, & respectively. Let e be the coefficient of mutual elasticity.

The velocity of A before impact may be resolved into two components, V cos a along 00' and Vsin a perpendicular to 00, while that of B may be resolved into v cos B and v sin ß in the same directions respectively. Now since the spheres are smooth, the whole action between them is in the line 00'. Hence the components of their velocities perpendicular to this line remain unaffected. We have therefore

Since action and reaction are equal and opposite, the impulse upon A is equal and opposite to that upon B; hence the change of A's momentum must be equal and opposite to that of the momentum of B, and the change in each takes place wholly in the direction of the impulse, that is, along the

line 00'. Hence equating the momentum lost by A to that gained by B, we have

M (V cos α- V, cos 0) = m (v, cos & — v cos B),

or MV, cos 0+mv, cos = MV cos a + mv cos ẞ......(II).

Also the velocity, in the direction of the impulse, of either sphere relative to the other after impact is to that before impact as e to 1. Hence

V, cos Ꮎ

1

· v1 cos &= − e (V cos a v cos ẞ)................(III).

The four equations (I), (II), and (III) completely determine V1, v1, 0 and 4 in terms of the given quantities. From equations (II) and (III) we obtain, precisely as in the preceding article,

MV cos a+mv cos B+ eM (V cos a — v cos B)

...(V).

M+ m

hence the components of the velocities of A and B along and perpendicular to 00' are known, and the values of V, and v1 can be at once written down. Again, from (I) and (IV),

v sin B (M + m)

....

v cos B)

..(VII).

tan &= MV cos a + mv cos ẞ − eM (V cos a —

Hence the direction of motion of each sphere after impact is found.

To find the impulsive pressure between the balls, we observe that the momentum of A in the direction 00' before the

impact is MV cos a, and this is changed by the impulse into MV, cos 0. The measure of the impulse is therefore

or, if this impulse be denoted by I, we have

128. In all the cases we have investigated the expression for the impulsive pressure involves the factor 1+e, and is therefore greater in the ratio of 1 + e to 1 than it would have been had e been zero, but all other circumstances the same. Thus, if in any case of collision, I' measure the impulse when the coefficient of elasticity is zero, and I be the measure of the impulse when, all other things being the same, the coefficient of elasticity is e, we have

I = I' (1 + e).

Now if we examine somewhat more closely into what takes place when two bodies strike one another, we find that, in the first place, each of them becomes compressed or indented, but if they are elastic they subsequently recover more or less completely their original form. If they are inelastic they remain indented, and move on together with a common velocity. Now up to the instant of greatest compression the action is the same whether the balls are elastic or inelastic, and therefore at this instant, even though the balls be elastic, they will be moving with a common velocity, and the change of momentum produced in either ball up to this instant will be the same as though they were inelastic. The impulse required to produce this change of momentum is sometimes called the "force of compression." We have denoted it by I'. Now in the case of elastic balls, after the compression, or indentation, has attained its maximum, the balls begin to recover their form. The parts which have been compressed consequently swell out against one another, and the force which they exert on one another serves to separate the balls.

This force is sometimes called the "force of restitution." The time taken by the balls to recover their form, and therefore the time during which this force acts is so short, that we are unable to measure it, and we are consequently compelled to estimate the force of restitution, like that of compression, by the whole momentum generated by it; in other words, to consider it as an impulse. Let this impulse be denoted by I". Then the whole change of momentum produced in either ball during the impact is that due to the force of compression, together with that due to the force of restitution, and is therefore numerically equal to I'+I". Hence

or the force of restitution is equal to e times the force of compression. If e equal unity or the elasticity be perfect, the forces of restitution and compression are equal.

In many treatises the coefficient of elasticity is defined. as the ratio of the force of restitution to that of compression, and it is stated as the result of experiment that this is constant for the same materials. It should, however, be borne in mind, that the element observed in experiments on this subject is not the forces which act during the collision, but the velocities of the balls before and after impact, and the measures of the forces of compression and restitution are subsequently deduced from the results of these observations by the help of the Second Law of Motion. It would therefore seem that the method adopted in the preceding Articles is the more natural way of treating the subject.

The coefficient of elasticity is sometimes called "the coefficient of restitution.”

If when two bodies impinge upon one another they be acted upon by some finite force, as, for example, gravity, then, since the time during which they remain in contact is so short that we cannot measure it, the effect produced by the