Physics - W.E.P. Question with Solution | TestHub

Let $s$ be the distance the block travels up the incline. The block starts with kinetic energy $\frac{1}{2}m{v}_0^2$ and comes to rest, so its final kinetic energy is $0$. The change in kinetic energy is $\Delta KE=0-\frac{1}{2}m{v}_0^2$. The forces doing work are gravity and friction. The normal force does no work.

Work done by gravity: $W_g=-mg(s\sin \theta )$ (negative as displacement is opposite to the component of gravity along the incline).

Work done by friction: The normal force is $N=mg\cos \theta$. The friction force is $f_k=\mu N=\mu mg\cos \theta$. Since friction opposes motion, $W_f=-f_{k}s=-\mu mg\cos \theta \cdot s$. By the Work-Energy Theorem, $W_g+W_f=\Delta KE$.$$-mgs\sin \theta -\mu mgs\cos \theta =-\frac{1}{2}m{v}_0^2$$ $$mgs(\sin \theta +\mu \cos \theta )=\frac{1}{2}m{v}_0^2$$ Solving for $s$: $$s=\frac{v_0^2}{2g(\sin \theta +\mu \cos \theta )}$$