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.)

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. (Assume radius of cross-section of tube is negligible in comparison to R.) (A) √2g (h+2R) (B) (5/2)R (C) √g (5R-2h) (D) √2g (2R-h)

Tardigrade Admin · Accepted Answer

For minimum v, velocity of ball at the topmost point will be zero. By conservation of energy, (1/2) mv2=mg(2R-h) v=√2g(2R-h)

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 $R$ .)

$2 g (h + 2 R)$

$\frac{5}{2} R$

$g (5 R - 2 h)$

$2 g (2 R - h)$

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

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.)

2g(h+2R)​

25​R

g(5R−2h)​

2g(2R−h)​