Question

A ball rolling on the top of a staircase of each step with height $H$ and width $W$, with an initial velocity $U$ will just hit the $n^{th}$ step. Then $n$ =

• A. $\frac{2U^2{H}^2}{gW}$
• B. $\frac{2U^2{H}^2}{g{W}^2}$
• C. $\frac{2U^{2}H}{g{W}^2}$
• D. $\frac{2U{H}^2}{W^2}$

Answer 1

Answer: (C)

$\frac{2U^{2}H}{g{W}^2}$

In $n^{th}$ step:
Height $=nH$
Range $=nW$
$u_h=U;u_v=0$ where $u_h$ is horizontal and $u_v$ is vertical velocity

In horizontal motion:
$nW=u_h\times t$
$t=\frac{nW}{u_h}$

In vertical motion:

$nH=0+\frac{1}{2}g{t}^2$
$\Rightarrow nH=\frac{1}{2}\times g\times {\left(\frac{nW}{U}\right)}^2$
$\Rightarrow \frac{2H{U}^2}{g{W}^2}=n$