Question

Two objects start their motion at some time. One is sliding down from rest on smooth fixed slope and other being thrown from point R at an angle $\theta$ if both get to point P at same instant and at same speed.than if $si{n}^2\theta =\frac{4n+\sqrt{2m}}{2L}$ then, $(n\times L)+m$

Answer 1

For particle on incline

$v=0+gsin\alpha t\dots \left(1\right)$

$\&v^2=0^2+2gsin\alpha \frac{R}{cos\alpha }\dots \left(2\right)$

For projectile
$t=\frac{2vsin\theta }{y}\dots \left(3\right)$

$\&vcos\theta =\frac{R}{t}\dots \left(4\right)$

By (1) & (4) in (2) $\Rightarrow \frac{cos\theta }{cos\alpha }=\frac{1}{2}...\left(5\right)$

By (1) & (3) $sin\theta sin\alpha =\frac{1}{2}-\left(6\right)$

By (5) & (6) $si{n}^2\theta =\frac{12+\sqrt{208}}{32}$