Question

A particle is projected up an inclined plane of inclination $\beta$ at an elevation $\alpha$ to the horizontal. Find the ratio between $\tan \alpha$ and $\tan \beta$, if the particle strikes the plane horizontally.

Answer 1

(a) $T=\frac{2u\sin (\alpha -\beta )}{g\cos \beta }$
Now we shall consider the motion of particle along $OA$.
(b) When the particle strikes the plane horizontally
In this case $t=\frac{2u\sin (\alpha -\beta )}{g\cos \beta }$
and $0=u\sin \alpha -gt$
$u\sin \alpha =g\left[\frac{2u\sin (\alpha -\beta }{g\cos \beta }\right]$
$2\tan \beta =\tan \alpha \Rightarrow \tan \alpha :\tan \beta =2$.