233-235] PROBLEMS ON IMPULSES. 265 



Fig. 68. 



Thus A moves off as stated provided there is no second impact between 

 A and C. The condition for this is ucos (<- 0)>w, 



\ e 



or -J (1 e} u' cos 2 6 + u' sin 2 6>~ u' t 



1 -\-G 



which leads to sin 6> (1 - e)/(l + e). 



II. A particle is projected with velocity V from the foot of a smooth fixed 

 plane of inclination 6 in a direction making an angle a with the horizon (a>0). 

 Find the condition that it may strike the plane n times striking it at right angles 

 at the nth impact, e being the coefficient of restitution between the plane and the 

 particle. 



Since the velocity parallel to the plane is unaltered by impact, the motion 

 of the particle parallel to the plane is determined by the same equation as if 

 there were no impacts, thus at the end of any interval t from the beginning 

 of the motion the velocity parallel to the plane is Fcos (a- 6} gt sin 6. 



Let t lt t<^ ... t n be the times of flight before the first impact, between the 

 first and second, and so on. Then ^ is given by 



Fj sin (a - 6) - \gt? cos 6 0, 



* and thus t 1 = 2Vsm(a-6)/gcoB0. The velocity perpendicular to the plane 

 at time ^ is Fsin(a-d)-^ 1 cosd or - Fsin(a-0). Immediately after the 

 impact the velocity at right angles to the plane becomes eFsin(a 6} away 

 from the plane. We thus find that t 2 = et l9 t 3 =et 2t .... 



Hence t 1 + t 2 + ... + t n =- -~ '- is the interval from the begin- 



ning of the motion till the nth impact. By supposition, at the end of 



