Question

Two balls of the same mass are released simultaneously from heights h and $2h$ from the ground level. The balls collide with the floor and stick to it. Then the velocity- time graph of centre of mass of the two balls is best represented by:

• A. Diagram
• B. Diagram
• C. Diagram
• D. Diagram

Answer 1

The correct option is C Diagram

Find speed of centre of mass before lower ball strike to ground. Given: $h_1=h,h_2=2h$

As both the balls are released simultaneously, at any instant before the lower balls reach the ground both have the same velocity

$v=gt$

$v_{CM}=\frac{m{v}_t+m{v}_t}{2m}=v_{t^{\prime }}$

$v_t$ being the instantaneous velocity.

$\therefore v$ vs $t$ graphs is a straight line.

Find speed of centre of mass after lower ball strike to ground. Just after the lower ball strikes ground and comes to rest:

$V_{cm}=\frac{m{v}_t}{2m}=\frac{v_t}{2}$

Hence, the velocity suddenly drops to half its value. Hence, graphs (a) and (b) are chosen.

Find acceleration of centre of mass.
After collision:

$a_{CM}=\frac{m\left(g\right)+m\left(0\right)}{m+m}=\frac{g}{2}$

i.e the slope (of $v_t$ curve) should decrease to half.

Hence, (b) is the best option