A body is projected up along an inclined plane from the bottom with speed${\text{V}}_1$ . If it reaches the bottom of the plane with a velocity${\text{V}}_{\text{2}}$ , find$\left({\text{v}}_{\text{1}}{\text{/v}}_{\text{2}}\right)\text{if}\theta$ is the angle of inclination with the horizontal and $\mu$ be the co-efficient of friction.

When a body is moving up along inclined plane

${\text{a}}_1=g\left(\sin \theta +\mu \cos \theta \right)$

${\text{v}}_{\text{1}}\text{=}\sqrt{{\text{2a}}_{\text{1}}\text{s}}$

$=\sqrt{2\text{g}\left(\sin \theta +\mu \cos \theta \right)}$

Case – II

When body is moving down along inclined plane

${\text{a}}_2=\text{g}\left(\sin \theta +\mu \cos \theta \right)$

${\text{v}}_{\text{2}}=\sqrt{2{\text{a}}_{\text{2}}\text{s}}$

$=\sqrt{2\text{g}\left(\sin \theta -\mu \cos \theta \right)}$

$\frac{{\text{v}}_{\text{1}}}{{\text{v}}_{\text{2}}}=\sqrt{\frac{2\text{g}\left(\sin \theta +\mu \cos \theta \right)}{2\text{g}\left(\sin \theta -\mu \cos \theta \right)}}$

$=\sqrt{\frac{\sin \theta +\mu \cos \theta }{\sin \theta -\mu \cos \theta }}$