A ball falls from a height h and rebounds aft | vedclass

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Question

(C) When a ball falls from a height $h$,it strikes the floor and rebounds to a height $h_1 = h e^2$.After the second impact,it rebounds to a height $h

Vedclass · Accepted Answer

$\left(\frac{1+e^2}{1-e^2}\right) h$

Vedclass · Answer

(C) When a ball falls from a height $h$,it strikes the floor and rebounds to a height $h_1 = h e^2$.After the second impact,it rebounds to a height $h_2 = h_1 e^2 = h e^4$.In general,the height reached after the $n$-th rebound is $h_n = h e^{2n}$.The total distance $H$ covered by the ball is the sum of the initial fall and the subsequent upward and downward paths for each rebound:$H = h + 2h_1 + 2h_2 + 2h_3 + \dots$$H = h + 2(h e^2 + h e^4 + h e^6 + \dots)$$H = h + 2h(e^2 + e^4 + e^6 + \dots)$The term in the bracket is an infinite geometric progression with first term $a = e^2$ and common ratio $r = e^2$. The sum is $S = \frac{a}{1-r} = \frac{e^2}{1-e^2}$.$H = h + 2h \left( \frac{e^2}{1-e^2} \right)$$H = h \left( 1 + \frac{2e^2}{1-e^2} \right)$$H = h \left( \frac{1 - e^2 + 2e^2}{1 - e^2} \right)$$H = h \left( \frac{1 + e^2}{1 - e^2} \right)$

$A$ ball falls from a height $h$ and rebounds after striking the floor. The coefficient of restitution is $e$. The total distance covered by the ball before it comes to rest is

Similar Questions

$A$ sphere of mass $m$ moving with velocity $v$ collides head-on with another sphere of the same mass which is at rest. The ratio of the final velocity of the second sphere to the initial velocity of the first sphere is (where $e$ is the coefficient of restitution and the collision is inelastic).

$A$ ball of mass $m$,moving with a speed $2v_0$,collides inelastically $(e > 0)$ with an identical ball at rest. Show that
$(a)$ For head-on collision,both the balls move forward.
$(b)$ For a general collision,the angle between the two velocities of scattered balls is less than $90^o$.

$A$ ball $P$ collides with another identical ball $Q$ at rest. For what value of the coefficient of restitution $e$ will the velocity of ball $Q$ become two times that of ball $P$ after the collision?

$A$ body falling from a height of $10\,m$ rebounds from a hard floor. If it loses $20\%$ of its energy in the impact,then the coefficient of restitution is:

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