ILLUSTRATIVE EXAMPLE 249 



The coin before this started into motion with a velocity V given by equa- 

 tion (a). Eliminating u from equations (a) and (c), we find as the relation 



between successive velocities of starting 



47/2 



a difference equation with constant coefficients. 



If e = 1, we notice from equation (6) that s = 0, so that each coin remains 

 absolutely at rest after striking the coin next in front of it it transmits the 

 whole of its momentum to this coin. Also from equation (a), 



After the momentum has been transmitted over the space between n coins, 

 the value of the square of the velocity is reduced by 2 n/tgrd. Thus at any point 

 the velocity of a moving coin is that which would be possessed by a coin which 

 had been started with velocity F, and made to move over a distance equal to all 

 the intervals between the coins over which the motion has been transmitted. 



If d = 0, so that the coins were originally in contact, we have 



V - 



1 + e 

 Thus, if there are n coins, the nth coin will start in motion with a velocity 



F 



EXAMPLES 



1. Hailstones are observed to strike the surface of a frozen lake in a direction 

 making an angle of 30 degrees with the vertical, and to rebound at an angle of 

 60 degrees. Assuming the contact to be smooth, find the coefficient of elasticity. 



2. If the hailstones of the last question rise after impact to a height of 2 feet, 

 find the velocity with which they originally struck the ground. 



3. In the last question find the height to which the hailstones will rise in 

 their second rebound from the ice. 



4. A ball is dropped on to a horizontal floor and, after rebounding twice, 

 reaches a height equal to half that from which it was dropped. Find the 

 coefficient of elasticity. 



"5. A bullet strikes a rough target at 45 degrees, and rebounds at the same 

 angle. Show that I e 



" = T+~e' 



6. A shot fired from a distance a strikes a target at right angles, and 

 rebounds. Show that it will fall at a distance ae from the target (neglecting 

 the resistance of the air). 



^ 7. A sphere of mass m collides with a sphere of mass ra' at rest, the contact 

 between them being smooth, and their paths after collision are observed to be 

 at right angles. Prove that m em'. 



