There are $n$ elastic balls placed on a smooth horizontal plane. The masses of the balls are $m, \frac{m}{2}, \frac{m}{2^2}, \ldots . \frac{m}{2^{n-1}}$ respectively. If the first ball hits the second ball with velocity $\mathrm{v}_0$, then the velocity of the $\mathrm{n}^{\text {th }}$ ball will be,

Question

There are $n$ elastic balls placed on a smooth horizontal plane. The masses of the balls are $m, \frac{m}{2}, \frac{m}{2^2}, \ldots . \frac{m}{2^{n-1}}$ respectively. If the first ball hits the second ball with velocity $\mathrm{v}_0$, then the velocity of the $\mathrm{n}^{\text {th }}$ ball will be,, $\frac{4}{3} v_0$, $\left(\frac{4}{3}\right)^n v_0$, $\left(\frac{4}{3}\right)^{n-1} v_0$, $v_0$

MARKS App · Accepted Answer

Hint : 1st Collision $\mathrm{V}_1=\frac{2 \times \mathrm{m}}{\frac{\mathrm{m}}{2}+\mathrm{m}} \mathrm{V}_0=\frac{4}{3} \mathrm{~V}_0$ 2nd Collision $V_2=\frac{2 \times \frac{m}{2}}{\frac{m}{2}+\frac{m}{2^2}} \times V_0=\left(\frac{4}{3}\right)^2 V_0$ 3rd Collision $\left(\frac{4}{3}\right)^3 \mathrm{~V}_0$ $\ldots(n-1)$ collision, $\therefore V_{n-1}=\left(\frac{4}{3}\right)^{n-1} V_0$

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