Question

A staircase contains three steps each $10cm$ high and $20cm$ wide (Figure). What should be the minimum horizontal velocity of a ball rolling off the uppermost plane so as to hit directly the lowest plane. ?

Answer 1

$h=$ height

$w=$ width of each step

$O=$ uppermost plane

$A=$ lowermost plane

Consider the ball roll from the upmost plane $O$ with horizontal velocity $v$ here the vertical velocity in the $y$ direction will be $0$

$X=vt$

$t=\frac{x}{v}...................\left(1\right)$

The horizontal range of the ball lies between $2wand3w$

Horizontal range $x$ of the ball will be greater than $2w$.

$\begin{array}{c}h=10cm=0.10m\\ w=20cm=0.20m\\ i.e,x\ge 2w\\ x\ge 3\times 0.20\\ x\ge 0.40m\end{array}$

The ball travels form plane $O$ to plan $A=2h$

equation of motion, $s=ut+\left(\frac{1}{2}\right)g{t}^2$

$u=0$ for vertical motion

$2h=ut+\left(\frac{1}{2}\right)g{t}^2$

$t=\sqrt{\frac{4h}{g}}......................\left(2\right)$

form (1)and (2)

$\frac{x}{v}=\sqrt{\frac{4h}{g}}$

$x=\sqrt[v]{v\frac{4h}{g}}$

we know $x\ge 0.40m$

$\sqrt[v]{v\frac{4h}{g}}\ge 0.40m$

$0.20v\ge 0.40$

$v\ge 2m/s$