244 MOTION OF SYSTEMS OF PABTICLES 



If, as before, we suppose that 6, <f> are the angles which the path 

 makes with the normal before and after the impact (see fig. 126), 

 we have 



tan 9 = - = 



v r ev 



so that e tan (/> = tan 6 (1 -f- e) p. 



The value of (1 4- e) //, is always positive, so that <f> is less than it 

 would be if the plane were smooth ; in other words, the roughness 

 of the plane causes the particle to rebound nearer to the normal. 



This equation, however, is only true within certain limits, for 

 we have assumed that there is sliding during the whole time of 

 impact. It may be tha^p. a certain stage of the motion sliding 

 will give place to rolling, and if so the equation we have obtained 

 is no longer valid. 



IMPACT OF Two MOVING BODIES 



201. Suppose now that two bodies A, B of masses m, m' im- 

 pinge at the point C, the common normal to C being the line CP. 

 Let it be supposed that the centers of 

 gravity of the two bodies both He in the 

 line CP at the moment of impact, and let 

 the components along CP of the velocities 

 of the centers of gravity of the masses 

 A, B respectively be 



u y u f before impact, 



V, V at the instant of greatest 



compression, 

 127 v, v r after impact. 



Then if we denote the impulse of compression by 7, and the 

 impulse of restitution by 7', we have 



-r , / -rr\ rm1 InJ V\ (?t 5^ 



7'= m(V- v) = - m'(V- v f ). (86) 



