PARTICLE IMPINGING ON A FIXED SURFACE 243 



The kinetic energy before impact is 

 %m(u 2 +v 2 ), 

 that after impact is ^ m (u 2 + v' 2 ). 



Thus there is a loss of kinetic energy of amount 



l m (^__^ 2) 

 or ^mv 2 (l e 2 ). 



This vanishes if e = 1, i.e. if the bodies are perfectly elastic. 

 In all other cases there is a loss of energy. We again see that e 

 cannot be greater than unity, or it would be possible to gain 

 energy by causing bodies to impinge on one another. 



Oblique Impact : Rough Contact 



200. As in the case of a smooth contact, we obtain the relation 

 v' = ev connecting the components of velocity along the normal. 

 The reaction, however, no longer acts entirely along the normal, so 

 that it is not now true that the tangential component of velocity 

 remains unaltered. 



Let us consider the case in which the surface of the particle 

 slides in the same direction over the fixed surface during the 

 whole time that the two surfaces are in contact. Then at every 

 instant of the impact there will be a tangential force equal to 

 //. times the normal force, so that the total tangential impulse 

 must be /z times the total normal impulse, and therefore equal 

 to /*(/+/'). 



Thus if u f is the tangential velocity after the impact, we have 



m(uu') = /*(/+/') 

 /*(!+)! 



= ft(l -\-e)mv, 

 so that u' = u ( 1 + e) pv. 



