(10) A ball A impinges on an equal ball B, which is at rest: supposing the velocities of the two balls after collision to be equal, determine the change of direction in the motion of A.

If e denote the elasticity of the balls, the angle between the directions of A's motion before and after collision is equal to tan‍1 (e*).

(11) A ball A impinges obliquely on another ball B, which is at rest, and, after collision, the directions of motion of A and B make equal angles with A's previous motion: find these angles.

If m, m', be the masses of A, B, respectively, e their mutual elasticity, and either of the required angles,

(12) An imperfectly elastic ball lies on a billiard-table: determine the direction in which an equal ball must strike it, in order that they may impinge upon a side of the table at equal given angles.

If e be the elasticity of the balls, a the magnitude of each of the given angles, fig. (127), and the inclination of the direction of the motion of the impinging ball to the line of centres of the balls at the instant of collision,

(13) A ball, moving on a smooth horizontal table, impinges on two others in all respects like it, which are lying in contact, at the same moment: determine the mutual elasticity of the balls, if the impinging one be brought to rest.

(14) Two balls, of equal volumes and masses, moving with equal velocities, in directions passing through the centre of a third ball, which is at rest, impinge upon it, and upon one another simultaneously: compare the mass of either of the impinging balls with that of the third ball in order that, after collision, the

directions of motion of the two impinging balls may be perpendicular to that of the third, the coefficient of elasticity being ·

1

The mass of the third ball must be four times as great as that of either of the impinging balls.

(15) Three equal and perfectly elastic balls A, B, C, move with equal velocities towards the same point, in directions equally inclined to each other; suppose, first, that they impinge upon each other at the same instant; secondly, that B and C impinge on each other, and immediately afterwards simultaneously on A; and, thirdly, that B and C'impinge simultaneously on A just before touching each other; and let V1, V2, V1, be the velocities of A after impact on these suppositions respectively prove that

CHAPTER VI.

IMPACT OF FALLING BODIES.

(1) FROM what height must a weight of 12 pounds fall in order that it may impinge upon the ground with the same effect as a weight of 25 pounds falling 6 feet?

The required height is 26 feet.

(2) Two inelastic bodies are dropped from P, Q, on to a smooth inclined line XY, and reach the lowest point of the line with the same velocity: prove that PQ is perpendicular to XY.

(3) Two perfectly elastic balls are dropped from two points not in the same vertical line, and strike against a perfectly elastic horizontal plane: shew that their centre of gravity will never reascend to its original height, unless the initial heights of the balls be in the ratio of two square whole numbers.

(4) Two equal perfectly elastic balls are dropped at the same

above a horizontal table: prove that, at the end of 6n+ 1 seconds, n being any positive integer, the velocity of the centre of gravity suddenly changes from g to 0, or from 0 to g.

If h' =

(3600)2
2p2

·9, p being any positive integer, prime to 2, 3,

and 5, prove that it will be exactly two hours before the centre of gravity attains its original altitude.

CHAPTER VII.

IMPACT OF PROJECTILES.

SECT. 1. Impact in the plane of motion.

(1) THERE are two parallel walls, the distance between which is equal to their height: from the top of one of them a perfectly elastic ball is thrown horizontally, so as to fall at the foot of the same wall, after rebounding from the other: to determine the position of the focus of the first path.

Let h be the height of each wall, u the velocity of projection, and t the time of flight between the instant of projection and the return of the ball to the foot of the first wall.

Then, the vertical descent not being affected by the impact at the second wall,

1 gt2

(1).

Again, the horizontal velocity being the same in magnitude throughout the motion, and h being the distance between the walls,

The equation (3) shews that the height of the directrix of the first path above the point of projection is equal to h: hence, the point of projection being the vertex of the parabola, the focus must be at the foot of the first wall.

(2) A heavy particle, of elasticity e, is projected, at an angle of inclination ẞ to a given inclined plane of altitude h, from the foot of the plane, so that after one rebound it just reaches the top of the plane, and describes part of the descending branch of the parabola after passing the top: to determine the latus rectum of the parabola.

Let V be the velocity of projection, and t, t, the times of the first and second bounds.

Then, since, by the nature of the problem, the velocity of the particle along the plane is zero at the top of the plane, we have Vcos ẞg sin a (t + t')

..(1).

........ .....

Again, since Vsin ß, eV sin B, are the velocities, normal to the plane, at the commencement of the first and second bound, respectively, we have also

Again, when the particle leaves the plane, its horizontal velo

but, since the particle only just reaches the top of the plane,

(3) A parabola is placed with its axis vertical and vertex downwards, and a perfectly elastic ball, dropped vertically, strikes the parabola with the velocity acquired in falling freely from rest through a distance equal to one-fourth of the latus rectum to find where it must strike the parabola, that after reflection it may pass through the vertex.

Let V be the velocity of impact, A the vertex, S the focus, and P the required point. Let t be the time between leaving P and arriving at A.

W. S.

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