266 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



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

 Fcos (a - 0) /^sin 6. The required condition is therefore 



tan 6 = 2 tan (a - 6} (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 + m'v' sin a = m'v sin a. 



By the generalised Newton's Kule we have 



Whence 



wsma= ev. 



m'sina(l+e) 



j-V-s-^v. 

 m+m sm 2 a 



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

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

 distances a, b from the two ends. C is projected with 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. 



Iv+at* 



Fig. 70. 



