20.] IMPACT OF SPHERES. 9 



which agrees with the result (9) found in Art. 15. For per : 

 fectly elastic bodies e=i, and formula (10) shows that in this 

 case the relative velocity after impact is numerically equal to 

 that before impact, but of opposite sense. 



20. Exercises. 



(1) Two balls of clay (<?=o) weighing 2 and 3 oz. move in the same 

 direction. The heavier ball impinges from behind upon the lighter ball 

 at the moment when the latter moves at the rate of 15 ft. per second. 

 If the velocity of the lighter ball is doubled by the impact, what was the 

 original velocity of the heavier ball ? 



(2) Two glass balls (<?=i) weighing i Ib. and 12 oz., respectively, 

 move in the same line with velocities of 5 and 4 ft. per second. What 

 are their velocities after impact (a) if their original velocities were of 

 the same sense, (<) if they were of opposite sense ? 



(3) A ball weighing 5 Ibs.,' while moving with a speed of 51 ft. per 

 second, overtakes a ball of 7 Ibs. moving in the same line at the rate of 

 40 ft. per second. If the coefficient of restitution be ^, what are the 

 velocities of the two balls after impact ? 



(4) With the data of Ex. (3), show that the velocities after impact 

 would be equal if the balls were perfectly inelastic, and that these veloci- 

 ties would differ more than in Ex. (3) if the balls were perfectly elastic. 



(5) Find the velocity with which an elastic ball rebounds from a 

 fixed surface after impinging upon it perpendicularly with a velocity u. 



(6) To determine the coefficient of restitution, a ball is dropped 

 from a height H on a fixed horizontal plate of the same material, and 

 the height of rebound h is measured. Show that e = ^Jh/H. 



(7) A ball is dropped from a height H~ 12 ft. on a fixed horizontal 

 plate. Find the height h to which it will rebound if e = f . 



(8) If not disturbed, the ball in Ex. (7) will continue to fall and 

 rebound alternately, (a) What height does it reach at the tenth re- 

 bound? (3) In what time does it come to rest? (c) What is the 

 whole space described? 



(9) A number of equal, perfectly elastic balls are placed in contact so 

 that their centres are in a straight line. An equal ball impinges with a 

 velocity u along this line on the first ball of the row. Show that the 



