Question

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 $R$.

• A. $\sqrt{g(R-h)}$
• B. $\sqrt{g(2R-h)}$
• C. $\sqrt{2g(2R-h)}$
• D. $\sqrt{2g}$

Answer 1

Answer: (C)

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

By using conservation of energy, from the initial point to the highest point

$\frac{1}{2}m{v}^2=mg(2R-h)+\frac{1}{2}m{V}^2$

for minimum V, V=0

$\Rightarrow \frac{1}{2}m{v}^2=mg(2R-h)\Rightarrow v=\sqrt{2g(2R-h)}$