The centres of the spheres 1, 2 and 3 lie on a single straight line. Sphere 1 is moving with an (initial)velocity $v_{1}$ directed along this line and hits sphere 2. Sphere 2, acquiring after collision a velocity $v_{2}$ , hitssphere 3. Both collisions are absolutely elastic . What must be the mass of sphere 2 (in kg) for the sphere 3 to acquire maximum velocity (The masses $m_{1} a n d m_{3}$ of spheres 1 and 3 are 9 kg & 1 kg respectively)?

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Solution

let $V_{1}, V_{2}$ be the velocities of the spheres 1 and2 after the first collision respectively. and let $V_{2}$ and V be the velocities of spheres 2 and 3 after the second collision. m be the mass of sphere 2. V is the velocity of sphere 3 after second collision.
for first collision equation of conservation of momentum and coefficient of restitution are:
$9 V_{1} = 9 V_{f 1} + m V_{2} . . . . . . . (1) V_{2} - V_{f 1} = V_{1}$ .......(2)\\ for\quad second\quad collision\quad equations\quad are: $m V_{2} = m V_{f 2} + 1 \times V . . . . . . . (3) V - V_{f 2} = V_{2} . . . . . . . (4) s o l v i n g t h e e q u a t i o n s t o g e t V i n t e r m s o f V_{1} a n d m V = 36 V_{1} \frac{m}{m^{2} + 10 m + 9} = 36 V_{1} \frac{1}{m + 10 + \frac{9}{m}} w h i c h g e t s m a x w h e n m = 3.$

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