Question

Ch
A ball bounces on a staircase, such that it strikes at each step at the same point (midpoint of step) and also bounces to same height
2
above each step as shown in the figure. Co-efficient of restitution between ball & steps is e = Then
3​

Answer 1

The initial height, (height from which the ball falls to step)

$\sf{\longrightarrow h_1=h+4}$

The final height, (height reached after bouncing off the step)

$\sf{\longrightarrow h_2=h}$

Then coefficient of restitution is given by,

$\sf{\longrightarrow e=\sqrt{\dfrac{h_2}{h_1}}}$

$\sf{\longrightarrow\underline{\underline{e=\sqrt{\dfrac{h}{h+4}}}}}$

$\sf{\longrightarrow\dfrac{h+4}{h}=\dfrac{1}{e^2}}$

By rule of dividendo,

$\sf{\longrightarrow\dfrac{4}{h}=\dfrac{1-e^2}{e^2}}$

$\longrightarrow\underline{\underline{\sf{h=\dfrac{4e^2}{1-e^2}}}}$

Answer 2

Answer:-

The drop height/initial height from which it falls

$\sf H_1 = h + l$

$\longrightarrow \sf H_1 = h + 4 \:m\quad \quad \dots (1)$

The bounce height/final height to which the ball rises after getting bounced,

$\sf H_2 = h$

$\longrightarrow \sf H_2 = 4\:m$

We know that,

$\sf COR = e = \sqrt{ \dfrac{Bounce\; height}{Drop\;height}}$

$\longrightarrow \sf e = \sqrt{\dfrac{H_2}{H_1}}$

$\longrightarrow \sf e^2 = \dfrac{h}{h + 4}$

$\longrightarrow \sf \dfrac{1}{e^2} = \dfrac{h + 4}{h}$

$\longrightarrow \sf \dfrac{1}{e^2} = 1 + \dfrac{4}{h}$

$\longrightarrow \sf \dfrac{1}{e^2} - 1 = \dfrac{4}{h}$

$\longrightarrow \sf \dfrac{1 - e^2}{e^2} = \dfrac{4}{h}$

$\longrightarrow \underline{\underline{\sf{\green{h = \dfrac{4e^2}{1 - e^2}}}}}$