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  4. 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. (Assume radius of cross-section of tube is negligible in comparison to R.)

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Q. 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 $R$.
(Assume radius of cross-section of tube is negligible in comparison to $R$.)Physics Question Image

AIIMSAIIMS 2018Work, Energy and Power

Solution:

For minimum v, velocity of ball at the topmost point will be zero.
By conservation of energy, $\frac{1}{2} mv^{2}=mg\left(2R-h\right)$
$v=\sqrt{2g\left(2R-h\right)}$