248 MOTION OF SYSTEMS OF PARTICLES 



If, as in the game of billiards, the spheres are of equal mass and 

 the second sphere is originally at rest, we take m = m', u' = 0, and 



obtain cot a' = - 1 (1 - e) cot a, 0' = 0. 



Thus B starts into motion along the line of centers, as it obviously 

 must since the forces which set it in motion act along this line. 



Since e is always less than unity and a is necessarily acute, cot a' 

 must be negative, so that a f will be obtuse. If e. = 1, then a 1 = 90. 

 Thus if the spheres were perfectly smooth and perfectly elastic, A 

 would move at right angles to the line of centers after impact ; its 

 motion would be the same as if it had impinged on a perfectly 

 smooth and inelastic plane. 



ILLUSTRATIVE EXAMPLE 



A row of similar coins is placed on a rough table, the coins being at equal dis- 

 tances apart and in a straight line. The first coin is projected along this line so as 

 to impinge directly on the second. Find the resulting motion. 



Let e be the coefficient of elasticity for an impact between the two coins, and 

 /a the coefficient of friction between the coins and the table. Let m be the mass 

 of each coin, and d the distance between the nearest points of two adjacent coins. 



The normal reaction between a coin and the table is mg, so that the f rictional 

 force opposing the coin's motion is /iragr, and the retardation produced is ng. 

 Thus if a coin is started from its original position with a velocity F, its velocity 

 on reaching the next coin is reduced to M, where 



F 2 - M 2 = 2 ngd. (a) 



We now have two coins of equal mass impinging with velocities w, 0. Their 

 velocities after impact, say u, w', are given by the equations 

 v v'= eu (Newton's law), 

 v + u'= u (conservation of momentum). 



Thus =-i-w(l-e), 



v'=iu(l + e). 



After impact the coin originally in motion has a velocity u, and is retarded 

 by a f rictional retardation ^g. It accordingly comes to rest, if it does not collide 

 again in the meantime, after a distance s given by 



, = - t (6) 



2flr Sg 



while the coin which has been started into motion sets off with a velocity 



(c) 



