(24) An elastic ball is projected from a point in a plane, which is inclined to the horizon at an angle a, the direction of projection making an angle ẞ with the upward direction of the inclined plane: at the end of one rebound it shoots through a horizontal bore at a point in the plane, just large enough to admit it prove that

cot ẞ=2(1+e) tan a + e cot a.

(25) OA, OB, are two planes inclined at angles a, B, to the horizon: a perfectly elastic ball is projected from A and strikes B, and continues to rebound along the same curvilinear path between A and B: having given the length of the chord AB, find its inclination to the horizon and the time of flight. If c be the length of the chord AB, y its inclination to the horizon, and t the time of flight, then

(26) From what height must a perfectly elastic ball be let fall into a fixed hemispherical bowl, in order that it may rebound horizontally at the first impact and strike the lowest point of the bowl at the second?

to

If be the radius of the bowl, the required height is equal

(27) A perfectly elastic ball is projected with a given velocity from a point between two parallel walls, and returns to the point of projection, after being once reflected at each wall: prove that its angle of projection is either of two complementary angles.

(28) A ball, of elasticity e, is projected from a point in a horizontal plane: if the distance of the point of nth impact be equal to four times the sum of the greatest altitudes of the ranges, determine the angle of projection,

(29) A ball is projected from a point A, in a horizontal line AB, at an elevation of 45° against a wall BC, and in a vertical plane perpendicular to the wall: after impact at C, it strikes the ground between A and B, and arrives at A after n rebounds find the ratio of BC to AB in terms of the elasticity of the ball, which is supposed to be the same in relation both to the wall and to the ground.

(30) A perfectly elastic ball is projected at an inclination B to a plane inclined to the horizon at an angle a, so as to ascend it by bounds: find the condition that the ball may rise vertically at the nth bound.

The required condition is expressed by the equation

cot a. cot ẞ= 2n + 1.

(31) A perfectly elastic ball is projected in a direction perpendicular to an inclined plane: prove that the ranges of the successive hops on the plane are in arithmetical progression, and that one straight line will be a tangent to all the curves described.

(32) A ball, of elasticity e, is projected with a velocity V at an angle ẞ to the upward direction of a plane inclined to the horizon at an angle a, and in a plane perpendicular to the intersection of the inclined plane with the horizon: prove that the ball will cease to hop at a distance a from the point of projection, provided that

(33) A ball, of elasticity e, is projected with a certain velocity at an angle ẞ to the upward direction of a plane inclined to the

horizon at an angle a: find the condition that the ball may arrive at the point of projection and cease hopping simultaneously. The required condition is expressed by the equation tan a. tan ẞ= 1 − e.

(34) A ball, of elasticity e, is projected from the foot of a smooth plane, inclined at an angle a to the horizon, in a direction making an angle ẞ with the upward direction of the plane: prove that it will in the nth bound move up or down the plane accordingly as

SECT. 2. Impact in any direction.

(1) An imperfectly elastic sphere is projected with a velocity V and impinges against a given vertical plane, which makes an angle ẞ with the plane of the sphere's motion, a being the distance of the point of projection from the given plane, and a the inclination of V's direction to the horizon: to find the velocity V' of the sphere just after impact; and to determine the values of a and V', when the elasticity is perfect, in order that V' may be a minimum.

If t be the time between the instants of projection and impact,

The normal velocity of impact is equal to Vcos a sin ß, and therefore the normal velocity of resilition is equal to

eV cos a sin B.

Again, the vertical velocity at the instant of impact is equal to

Also, the horizontal velocity, parallel to the given vertical plane, at the instant of impact, is equal to Vcos a cos B.

From this result we see that V' is least when

the corresponding value of V' being given by the equation

(2) A perfectly elastic ball is projected with a given velocity from the middle point of one of the sides of a given equilateral three-cornered room: it strikes the two other sides and returns to the point of projection: to determine the direction in which it was projected.

Let ABC, fig. (129), be the floor of the room, PQR the projection of the path of the ball upon the floor, P being that of the point whence the ball started.

It is easily seen that the angles indicated in the figure by are equal, and that those indicated by o are also equal.

Hence, as is easily seen, the triangle PQR is equilateral, each side being equal to half a side of ABC.

Let V = the velocity of projection, a = the inclination of the direction of projection to the horizon, and a = the length of a side of ABC.

Since the lines PQ, QR, RP, are all equal, the times through the arcs, of which they are the projections, must be equal: let t denote the time through each.

Then, Vsin a being the vertical component of the velocity of projection and t the whole time of flight,

and, a being the length of each side of the triangle PQR,

(3) Three planes, two vertical and one horizontal, are mutually at right angles: a ball is projected from a given point in the horizontal plane, with a given velocity and in a given direction, and strikes successively each of the two vertical planes: to find the point at which it will again strike the horizontal plane, the coefficient of elasticity being supposed to be the same between the ball and each of the vertical planes.

Let OA, OB, OC, fig. (130) be the intersections of the three planes, OC being vertical, and OA, OB, horizontal: let E