n elastic balls are placed at rest on a smooth horizontal plane which is circular at the ends with radius r as shown in the figure. The masses of the balls are m, m2, m22, .....,m22-1 respectively. What is the minimum velocity which should be imparted to the first ball of mass m such that this nth ball will complete the vertical circle?

In head-on elastic collision v1'=m/2−mm/2+m(0)+2(m)m/2+m(v)=43vFinally, v'n=43n−1⋅v=5gr ∴ v=34n−15gr

n elastic balls are placed at rest on a smooth horizontal plane which is circular at the ends with radius r as shown in the figure. The masses of the balls are m, m2, m22, .....,m22-1 respectively. What is the minimum velocity which should be imparted to the first ball of mass m such that this nth ball will complete the vertical circle?

In head-on elastic collision v1'=m/2−mm/2+m(0)+2(m)m/2+m(v)=43vFinally, v'n=43n−1⋅v=5gr ∴ v=34n−15gr

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n elastic balls are placed at rest on a smooth horizontal plane which is circular at the ends with radius r as shown in the figure. The masses of the balls are $m, \frac{m}{2}, \frac{m}{2^{2}}, . . . . ., \frac{m}{2^{2 - 1}}$ respectively. What is the minimum velocity which should be imparted to the first ball of mass m such that this nth ball will complete the vertical circle?

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${(\frac{3}{4})}^{n - 1} \sqrt{5 g r}$

${(\frac{4}{3})}^{n - 1} \sqrt{5 g r}$

${(\frac{3}{2})}^{n - 1} \sqrt{5 g r}$

${(\frac{2}{3})}^{n - 1} \sqrt{5 g r}$

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Detailed Solution

In head-on elastic collision

$v_{1}^{'} = (\frac{m / 2 - m}{m / 2 + m}) (0) + [\frac{2 (m)}{m / 2 + m}] (v) = \frac{4}{3} v$

Finally, $v'_{n} = {(\frac{4}{3})}^{n - 1} \cdot v = \sqrt{5 g r} ∴ v = {(\frac{3}{4})}^{n - 1} \sqrt{5 g r}$

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