an equal ball A impinge on B in the direction of the straight line. By this collision A is brought to rest, and B proceeds, with the velocity which A had, to strike C; then B is brought to rest and C proceeds to strike D; finally C is brought to rest, and D proceeds with the velocity which à originally had. If the balls B, C, D were at first close together it is curious to see D fly off apparently immediately A strikes B. Ivory balls, though not perfectly elastic, are sufficiently elastic to exhibit the experiment well.

279. Next suppose that there is a collision between two balls which are inelastic. Then after collision the balls do not separate, but move on together with the same velocity. This velocity can be determined when we know the velocities of the balls before the collision by the aid of the principle that the momentum of the system is the same after collision as before. The principle can be expressed briefly in the foregoing words; but a little explanation is necessary in order to fix the meaning of the term momentum of the system. If the balls are moving in the same direction the momentum of the system is the sum obtained by adding the momentum of one ball to the momentum of the other; if the balls are moving in opposite directions the momentum of the system is the difference obtained by subtracting the momentum of one ball from the momentum of the other. If one body is at rest before the collision then the momentum of the system is the momentum of the other body. This principle of the identity of the momentum of the system before and after collision is shewn by theory to be an obvious consequence of the Third Law of Motion.

280. Suppose for example that a ball of mass 5 moving with a velocity 6 strikes a ball of mass 4 moving with a velocity 3; the velocities may be understood to be expressed throughout in feet per second. Then if the balls are moving in the same direction the momentum of the system before impact is 30+12, that is 42. After the collision the balls move on together with the velocity

42

9

; for with this velocity, since the whole mass is 9, the

momentum of the system after the collision will be 42, the same as before. But if the balls are originally moving in contrary directions the momentum of the system before the collision is 30-12, that is 18; and so after

18

collision the balls move on with the velocity that is 9 9

2: the direction of this velocity is the same as that of the ball which had the greater momentum before the collision.

281. The general problems of the direct collision of elastic balls which are not equal in mass, and of imperfectly elastic balls, do not yield results which we can express simply and easily in words; except that the principle just stated with respect to momentum always holds, namely that the momentum of the system is the same after the collision as before. But numerical results are easily obtained by following the steps of Art. 272. Suppose for example that a ball of mass 5 moving with a velocity 6 overtakes a ball of mass 4 moving with a velocity 3. If the balls are inelastic we found in Art. 280 that they would move

42

on together with the velocity that is 14. Suppose

9

3

however that instead of being inelastic the balls have

14

3

4

for their index of elasticity; then will still be the com

3

mon velocity at the end of the first part of the impact. Thus the ball which had originally the velocity 6 has lost

4

that is ; and therefore during the second part of

originally the velocity 3 gained during the first part of the

282. We have hitherto supposed that both balls are in motion; or at least if one ball is at rest before collision we have supposed that it is moveable. But a particular case may be noticed of another kind, namely that in which one body moves and strikes another which is fixed; we may for simplicity take this fixed body to be a plane. We suppose the collision to be direct. Then it is found that if the moving ball and the fixed plane are inelastic the moving ball remains close to the fixed plane after collision; and if the moving ball and the fixed plane are perfectly elastic the moving ball recoils after collision in the same straight line and with the same velocity as before. If the moving ball and the fixed plane are imperfectly elastic the moving ball recoils after collision in the same straight line as before, with a velocity which is equal to the product of the former velocity into the index of elasticity.

283. The next subject which naturally occurs for consideration is the oblique collision of bodies.

In the diagram let C represent the centre of one ball, and CA the direction in which it is moving at the instant

B

of collision; let Crepresent the centre of the other ball, and C'A' the direction in which it is moving at the instant of collision. Then it is found by theory that we may treat the problem thus. Resolve the velocity of the ball whose centre is C into two components, one along CC', and the other at

T. P.

8

right angles to CC'; also resolve the velocity of the ball whose centre is C into two components, one along CC', and the other at right angles to CC. Then the velocities along CC are changed in precisely the same way as if the balls moving with these alone came into direct collision; and the velocities at right angles to CC are not affected at all; that is they remain the same for each ball after collision as before. Since we thus know the two component velocities of the ball whose centre is C, we can find the resultant velocity after collision, and the direction, CB, of this velocity. Similarly we can find the resultant velocity after collision of the other ball, and its direction C'B'.

It is often convenient to resolve velocities into components in the manner just exemplified; the method is the same as for resolving forces: see Art. 156.

284. An important case of oblique collision is that in which a moving ball strikes a fixed plane.

Let AC represent the direction in which the ball moves before it strikes the fixed plane at C; let CD be at right angles to the plane. After striking the plane the ball will go off in some direction which we denote by CB. The angle ACD is called the

D

C

B

angle of incidence, and the angle BCD the angle of reflection. If the ball and the fixed plane are perfectly elastic these angles are equal, and the velocity of the ball after collision is equal to the velocity before. If the ball and the fixed plane are imperfectly elastic the angle of reflection is greater than the angle of incidence, the relation between the two depending on the index of elasticity. In the case in which the ball and the fixed plane are inelastic the angle of reflection is a right angle, so that the ball after collision moves close to the plane. The velocity after collision is always less than the velocity before collision, except when the ball and the fixed plane are perfectly elastic.

285. Many remarkable results are obtained by the collision of balls on a billiard-table, which the principles we have stated would not be sufficient to explain. These results depend on two circumstances which we have not considered, namely the rotation of the balls, and the friction between the balls, and between the balls and the table: the theory of such results would be altogether beyond the present work.

XIX. MOTION DOWN AN INCLINED PLANE.

286. We have already spoken about the motion of a body falling freely, but we will now make a few additional remarks on the subject. The motion in this case is said to be uniformly accelerated: this means that in successive equal intervals of time the velocity of the falling body receives equal additions. The laws of the motion involve, as we saw, two numbers, namely 16 which expresses the number of feet fallen through in the first second of time, and 32 which expresses in feet per second the velocity at the end of the first second. The first number is half the second, and the reason for this may be seen without difficulty. The velocity increases in the same proportion as the time, and in the first second the velocity begins with the value 0 and ends with the value 32. Hence 16 may be called the average velocity; for instance at the end of the first tenth of a second the body is falling with the

1

velocity of 32, and at the end of nine-tenths of the

10

9

second it is falling with the velocity of 32: the sum of

10

these two velocities is 32, so that the half sum is 16. It is easy to admit that when the velocity increases or decreases uniformly as the time increases, then the space described in a given time is just the same as would be described by a body moving during that time uniformly with the average velocity, that is with a velocity equal to half the sum of the velocities at the beginning and the end of the given time.