EXAMPLES 253 



20. Prove that an elastic ball, let fall vertically from a height of h feet 

 on a hard pavement, and rebounding each time vertically with e times the 

 striking velocity, will have described 



l +e /2ft 



- - \l 

 1 e \ 



k feet, in - - \l seconds 



I e 2 

 before the rebounds cease. 



Work this out for h = 1, e = |. 



21. A ball is dropped from the top of a tower, height A, and at the same 

 time another ball of equal weight is projected upwards with the velocity 

 \/2gh from the base of the tower and collides directly with the falling ball. 

 If the coefficient of restitution be e, prove that the falling ball will, in 



the rebound, rise to a height short of the top of the tower by - (1 e 2 ). 



22. A boy standing on a railway bridge lets a ball fall on the horizontal 

 roof of a car passing under the bridge at 15 miles an hour. If /* = 1, e = * 

 between the roof and the ball, find the least height of the boy's hand above 

 the roof in order that the second rebound of the ball may be from the 

 same, point "of the roof as the first. 



If the boy's hand is at a greater height than this, what will happen? 



23. A perfectly elastic particle is projected so as to strike the inside of 

 a surface of revolution of which the axis is a given vertical line. Show 

 that the vertices of all the parabolas described after successive rebounds 

 lie on a surface of which the shape is independent of that of the surface 

 of revolution. 



24. Prove that in order to produce the greatest possible deviation in the 

 direction of motion of a smooth billiard ball of diameter a by impact on 

 another equal ball at rest, the former must be projected in a direction 

 making an angle 



with the line, of length c, joining the two centers. 



25. A pendulum hangs with its bob just in contact with a smooth verti- 

 cal plane. The bob is drawn aside until it is 5 inches higher than it was, 

 and is then released so as to strike the plane normally ; and on the first 

 rebound it rises vertically through 4 inches. What would have been the 

 vertical rise on rebound if the pendulum had been drawn aside through 

 the same angle, but so that the bob strikes at an angle of 60 degrees with 

 the normal? 



