Question

A block of mass $m$ slides without friction down a fixed inclined board of inclination $\alpha$ with the horizontal. After leaving the inclined, the block falls on a cart of mass $M$ filled with sand as shown.
Initial height of the block above the level of the cart is $h$ as shown. The velocity of cart when block drops on it will be

(Assume that block and cart move together, when block drops on to the cart)

• A. $\frac{m\sqrt{2gh}}{M+m}$
• B. $\frac{m\sqrt{2gh}\sin \alpha }{M+m}$
• C. $\frac{m\sqrt{2gh}\cos \alpha }{M+m}$
• D. $\frac{M\sqrt{2gh}\cos \alpha }{M+m}$

Answer 1

The correct option is (D) $\frac{m\sqrt{2gh}\cos \alpha }{M+m}$

$byusing3rdeqofmotion$

$v^2-u^2=2gh$

$v=\sqrt{2gh}$

Speed of block at the bottom of board is $=\sqrt{2gh}$

Let common velocity of cart and block is $v$

Applying conservation of linear momentum in horizontal direction,

$m\sqrt{2gh}\cos \alpha =(M+m)v$
$v=\frac{m\sqrt{2gh}\cos \alpha }{M+m}$
$\therefore \left(c\right)$