A block is sliding down on an inclined plane of slope $ heta$ and at an instant $t=0$ this block is

Question

A block is sliding down on an inclined plane of slope $	heta$ and at an instant $t=0$ this block is given an upward momentum so that it starts moving up on the inclined surface with velocity $u$. The distance ( $S$ ) travelled by the block before its velocity become zero, is $\_\_\_\_$ .
(g = gravitational acceleration)

Professor PC Thomas Classes · Answer

Since the block was initially sliding down the incline at a constant velocity (implied by typical problem phrasing of this type), the kinetic friction balances the parallel gravitational component, meaning $\mu \cos	heta = \sin	heta$. When projected upwards, both gravity and friction act downwards along the incline, producing a retardation $a = g\sin	heta + \mu g\cos	heta = 2g\sin	heta$. Using the kinematic equation $v^2 = u^2 - 2aS$ with $v=0$, the distance is $S = \frac{u^2}{2a} = \frac{u^2}{4g\sin	heta}$.