264 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



It follows that the statement W eW, which is the 

 generalised Newton's rule, is equivalent to the statement that 

 the kinetic energy lost is proportional to the square of the 

 relative velocity of approach, and, if this could be assumed, the 

 generalised Newton's rule could be deduced. This rule would 

 thus be obtained from an hypothesis as to the amount of energy 

 dissipated instead of an hypothesis as to the impulsive pressure. 



234. General theory of impulsive changes of motion. 



So far we have been confining our attention to the impulsive action 

 between impinging bodies, but there are many other changes of 

 motion which take place so rapidly that it is convenient to regard 

 them as impulsively produced. The general method of treating 

 such changes of motion has been indicated in Articles 82 and 113 ; 

 it depends simply on repeated applications of the statement that 

 for every particle in a connected system, and for each rigid body in 

 such a system, the changes of momentum are a system of vectors 

 equivalent to the impulses that produce them. We shall illustrate 

 the application of this statement in a number of problems. 



235. Illustrative problems. 



I. Two equal smooth balls, whose centres are A and B, lie nearly in 

 contact on a smooth table, and a third ball of equal size and mass impinges 

 directly on A, so that the line joining its centre C to A makes with the line AB 

 an angle CAB, IT 6. Prove that, if sin 6>(l-e)l(I+e'), the ball A will 

 start of in a direction making with AB an angle tan~ l {2 tan 6 /(I e)}, e being 

 the coefficient of restitution for either pair of balls. 



Let V be the velocity of C before striking A ; since the impact is direct, 

 F is localised in CA. Let w be its velocity after striking A ; the direction 

 of w is that of F. Let u' be the velocity of A immediately after C strikes it, u its 

 velocity just after A strikes B, v the velocity of B after A strikes it, then the 

 direction of u' makes an angle 6 with AB. Suppose that the direction of u 

 makes an angle < with AB. The direction of v is AB. 



We have the equations of momentum 



V=u'+w, u' cos0 = ucos<l>+v, 

 and the equations given by Newton's Rule 



u' w=eV, ucos<j) v 

 whence 2w=V(I-e), 2w'=F(l + e), 2wcos$ = (l-e) 'cos 0, 



2 tan# 

 and tan < = . 



