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With what minimum speed $v$ must a small ball should be pushed inside a smooth vertical tube from a height $h$ so that it may reach the top of the tube? Radius of the tube is $R$ .

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$\sqrt{4 g (3 R - h)}$

$\sqrt{6 g (2 R - h)}$

$\sqrt{8 g (2 R - h)}$

$\sqrt{2 g (2 R - h)}$

answer is D.

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Detailed Solution

$v_{top} = \sqrt{v^{2} - 2 g (2 R - h)}$

To just complete the vertical circle $v_{top}$ may be zero.

$0 = \sqrt{v^{2} - 2 g (2 R - h)} v = \sqrt{2 g (2 R - h)}$

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