235, 236] EXAMPLES OF IMPULSES. 267 



Since the impulse on C is along the string its direction of motion is unaltered. 

 The velocity with which P starts to move is v along the string. 



Let o&amp;gt; be the angular velocity with which the rod begins to turn. The 

 velocity of A is compounded of the velocity of P and the velocity of A 

 relative to P. Thus A starts with velocity v+aco. So B starts with velocity 

 v-ba&amp;gt;. 



The equation of linear momentum parallel to the string is 



mv + m ( v + aa&amp;gt;) + m (v - &o&amp;gt;) = mu, 

 m being the mass of either particle. 



The equation of moment of momentum about P is 



ma (v + o&amp;gt;) mb (v &&&amp;gt;) = 0, 

 giving a&amp;gt; = (b-a)v 



Eliminating w we find 



236. Examples. 



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



1. The sides of a rectangular billiard table are of lengths a and b. If a 

 ball is projected from a point on one of the sides of length b to strike all 

 four sides in succession and continually retrace its path, show that the angle 

 of projection B with the side is given by ae cot 6 = c + ec , where c and c are 

 the parts into which the side is divided at the point of projection. 



2. Prove that in order to produce the greatest deviation in the direction 

 of a smooth billiard ball of diameter a by impact on another equal ball at 

 rest, the former must be projected in a direction making an angle 



with the line (of length c) joining the two centres. 



3. A particle is projected from a point at the foot of one of two smooth 

 parallel vertical walls so that after three reflexions it may return to the 

 point of projection, and the last impact is direct. Prove that e s + e 2 + e = I, 

 and that the vertical heights of the three points of impact are in the ratios 



e 2 : 1 e 2 : 1. 



4. A particle is projected from the foot of an inclined plane and returns 

 to the point of projection after several rebounds, one of which is at right 

 angles to the plane ; prove that, if it takes r more leaps in coming down than 

 in going up, the inclination 6 of the plane and the angle of projection a are 

 connected by the equation 



cot 6 cot (a - 0) = 2 V(l - O - (1 - e r )}/{e r (1 - e}}. 



