266 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



this interval the velocity parallel to the plane vanishes, or this interval is 

 Fcos (a - &}lg sin 6. The required condition is therefore 



tan 6 = 2 tan (a - 0) (1 - e n )/(l - e). 



III. A smooth sphere of mass m, is tied to a fixed point by an inextensible 

 thread, and another sphere of mass m impinges directly on it with velocity v in 

 a direction making an acute angle a with the thread. Find the velocity with 

 which m begins to move. 



The impulse between the spheres acts in the line of centres so that the 



direction of motion of m is unaltered. 

 Let its velocity after impact be v . 



There is an impulsive tension in the 

 thread and the sphere m is constrained 

 to describe a circle about the fixed end. 

 It therefore starts to move at right 

 angles to the thread. Let u be its 

 velocity. 



Kesolving for the system at right 

 angles to the thread we have the equa 

 tion of momentum 



mu -f- m v sin a = m v sin a. 



Fig. 69. 



By the generalised Newton s Kule we have 

 v u sin a = ev. 



Whence 



sin a 



m+m sin 2 a 



IV. Two particles A, B of equal mass are connected by a rigid rod of 

 negligible mass, and a third equal particle G is tied to a point P of the rod at 

 distances a, b from the two ends. C is projected ivith velocity u perpendicular 

 to AB. Find the velocity of C immediately after the string becomes tight. 



Let v be the velocity of C immediately after the string becomes tight. 



AV-lw 



v+att 



Fig. 70. 



