A small ball of mass m is connected by an inextensible massless string of length l with another ball of mass M $$=$$ $$4m$$. They are released at $$t=0$$ with zero tension in the string from a height h as shown in the figure. The string becomes taut for the first time at $$t=lngh$$ after the mass M collides with the ground (Assume$$ all collisions to be $$elastic) . Find value of n .

Question

A small ball of mass m is connected by an inextensible massless string of length l with another ball of mass M $$=$$ $$4m$$. They are released at $$t=0$$ with zero tension in the string from a height h as shown in the figure. The string becomes taut for the first time at $$t=lngh$$ after the mass M collides with the ground (Assume$$ all collisions to be $$elastic) . Find value of n .

Infinity Learn · Accepted Answer

Before collision, the velocity of each ball is $V = \sqrt{2 g h}$ in the same direction.
After collision of M with ground, the velocity of M ball is $V = \sqrt{2 g h}$ in the opposite direction.

Relative acceleration between the two balls is zero and relative velocity of approach is $V - (- V) = 2 V = 2 \sqrt{2 g h}$ .
$∴$ The two balls will collide after a time $= \frac{r e l a t i v e d i s \tan c e}{r e l a t i v e v e l o c i t y} = \frac{l}{2 \sqrt{2 g h}}$ .
Since the collision is elastic, the velocity of separation is $2 \sqrt{2 g h}$
$∴$ The string becomes taut after same time $t = \frac{l}{2 \sqrt{2 g h}}$
$∴$ Total time will be 2t = $\frac{l}{\sqrt{2 g h}}$

Collisions

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