Question

Two balls $a$ and $b$ of same mass are moving on a circular path of radius $r=1\text{m}$ with constant velocity of $5\text{m/s}$ and $15\text{m/s}$ as shown in figure. If after some time both ball collides perfectly elastically, find the time $t$ when they will collide again. Friction is absent everywhere.

• A. $\frac{\pi }{5}\text{s}$
• B. $\frac{2\pi }{5}\text{s}$
• C. $\frac{\pi }{10}\text{s}$
• D. $\pi \text{s}$

Answer 1

The correct option is A $\frac{\pi }{5}\text{s}$

As we know after perfectly elastic collision of equal mass object, the velocities get interchanged. Therefore after $1st$ collision ball $a$ has velocity of $5\text{m/s}$ and ball $b$ has velocity of $15\text{m/s}$


Now, we will find time $t$ when both ball collide again with help of concept of relative motion.
After first collision, velocity of ball $b$ with respect to ball $v_{b,a}=v_b-v_a=(15-5)\text{m/s}=10\text{m/s}$


To collide again after time $t$, ball $b$ has to travel distance equal to perimeter of circle$\left(2\pi r\right)$ with respect to ball $a$.
$\therefore t=\frac{2\pi r}{v_{b,a}}=\frac{2\times \pi \times 1}{10}=\frac{\pi }{5}\text{s}$