A ball of mass `M//2` filled with a gas ( whose mass is `M//2`) is kept on a frictionless table . A bullet of mass `m = M//4` and velocity `v_(0)hat(i)` penetrates the ball , and rests inside at t=0. Assume that the amount of gas emitted during the collision can be neglected. The compressed gas is emitted at a contant velocity `v_(0)//2` relative to the ball and at an even rate k ( k is a positive constant ) .
(a) What is the velocity of the ball after the collision with the bullet ?
(b) Find the velocity of the ball `v(t)` as a function of time. Assume that the emission of gas starts at t=0. What is the final velocity of the ball ?

Answer

(a) `(v_(0))/( 5) hat( i ) `, ( b ) ` v_(0) [ ( 1)/( 5) + ( 1)/( 2) ln ((m+M)/(m + M - kt)) ] hat( i ) `, ( c ) 4.55 `v_(0)`