From a tower of height $H$, a particle is thrown vertically upwards with a speed $u$. The time taken

Question

From a tower of height $H$, a particle is thrown vertically upwards with a speed $u$. The time taken by the partcle, to hit the ground, is $n$ thmes that taken by it to reach the highest point of its path. The relation between $H, u$ and $n$ is

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(d) : Time taken by the particle to reach the top most point is, 
$t=\frac{u}{g}$ ...(i)
 Time taken by the particle to reach the ground $=n t$ Using, $s=u t+\frac{1}{2} a t^2$ \ $\Rightarrow \quad-H=u(n t)-\frac{1}{2} g(n t)^2$\ $\Rightarrow \quad-H=u 	imes n\left(\frac{u}{g}ight)-\frac{1}{2} g n^2\left(\frac{u}{g}ight)^2 \quad[$ using (i) $]$\ $\Rightarrow \quad-2 g H=2 n u^2-n^2 u^2 \Rightarrow 2 g H=n u^2(n-2)$