Question

Two balls marked 1 and 2 of the same mass m and a third ball marked 3 of mass M are arranged over a smooth horizontal surface as shown in the figure. Ball 1 moves with a velocity $v_1$ towards balls 2 and 3. All collisions are assumed to be elastic. If M < m, the number of collisions between the balls will be

• A. One
• B. Two
• C. Three
• D. Four

Answer 1

The correct option is B

Two

The first collision will be between balls 1 and 2. Since both have the same mass, after the collision ball 1 will come to rest and ball 2 will move with speed $v_1$. This ball will collide with the stationary ball 3. After this second collision, let $v_2$ and $v_3$ be the speeds of balls 2 and 3 respectively. Since the collisions are elastic, $v_2$ and $v_3$ are given by
$v_2=\left(\frac{m-M}{m+M}\right)v_1$ ...... (i)
and $v_3=\left(\frac{2m}{m+M}\right)v_1$ ...... (ii)
If M < m, it follows from (i) and (ii) that $v_2<v_3$ and both have the same direction. Therefore, ball 2 cannot collide with ball 3 again. Hence, there are only two collisions and the correct choice is (b).