240-242] COLLISION OF ROUGH BODIES. 273 



Writing F for the impulsive friction at the point of contact, and taking 

 the same notation as in the last Article, we have the equations of impulsive 

 motion 



(u-U)=-R, m(v-V)=-F,\ 



and 



m'(u'-U') = R, m'(v'-V') = F,\ 

 m't(*'-a') = (S-ar)F-to-y>)RS ' 



Also we have the equation of sliding friction 



F=f-R ....................................... (3), 



and the equation provided by the generalised Newton's Rule 



t*-fo-y)-tt' + ' (?-/)= -e{U-Q(r]-y}-U' + Q' (;-/)}. ..(4). 



From these equations we obtain, by elimination of u, u', v, v', o>, <>', F, an 

 equation for R, viz. 



[U-Q fo-y)- 7' + 0'fo-y)], 

 showing that R contains (1 + e) as a factor and is otherwise independent of e, 

 and thus proving the equivalence of the two rules. 



Further we can show that, when there is sufficient friction to prevent 

 sliding, the rules are not in general equivalent. 



We shall assume the generalised Newton's rule as a basis of discussion. 

 We have not in this case any relation between F and R, but equations (1), 

 (2), (4) still hold, with the additional equation of no sliding 



f + o)(-.*0 = 'y' + a>'(-O ........................... (5). 



From equations (1), (2), (4), (5) we can form two equations for R and 

 F, viz. 



to -^ W-*ni-y) . (J-aQOy-yri 



J 



. 



rn 



r^~^^~^ 4. tf-^^-yoi 



L ^ 2 ^^ 2 J 



= 7+0 (-*)- F'-Q'd-^). 



It is clear that the solution of these equations will give an expression for R 

 consisting of two terms, one of them having (1+e) as a factor and the other 

 not containing that factor. 



*242. Examples. 



[In these examples e is the coefficient of restitution between two bodies.] 

 1. From the rule deduced from Poisson's hypothesis obtain the general- 

 ised Newton's rule for smooth bodies, and for bodies with sliding friction. 

 L. 18 



