A-ball-falls-vertically-on-to-a-floor-with-momentum-p-and-then-bounces-repeatedly-the-coefficient-of-restitution-is-e-The-total-momentum-imparted-by-the-ball-to-the-floor-is-

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Question Number 23556 by Tinkutara last updated on 01/Nov/17

$$\mathrm{A}\mathrm{ball}\mathrm{falls}\mathrm{vertically}\mathrm{on}\mathrm{to}\mathrm{a}\mathrm{floor},\mathrm{with}$$$$\mathrm{momentum}p,\mathrm{and}\mathrm{then}\mathrm{bounces}$$$$\mathrm{repeatedly},\mathrm{the}\mathrm{coefficient}\mathrm{of}\mathrm{restitution}$$$$\mathrm{is}e.\mathrm{The}\mathrm{total}\mathrm{momentum}\mathrm{imparted}\mathrm{by}$$$$\mathrm{the}\mathrm{ball}\mathrm{to}\mathrm{the}\mathrm{floor}\mathrm{is}$$

Answered by ajfour last updated on 01/Nov/17

$$J_1=m({ev}_0+v_0)$$$$J_2=m(e^2{v}_0+{ev}_0)$$$$..$$$$..$$$$\underset{i=1}{\sum ^{\mathrm{\infty }}}{J}_i={mv}_0(1+e)[1+e+e^2+\dots ]$$$$=\left(\frac{1+e}{1+e}\right)p.$$

Commented by Tinkutara last updated on 01/Nov/17

$$\mathrm{But}\mathrm{why}\mathrm{not}{J}_1={mv}_0\mathrm{only}?$$

Commented by ajfour last updated on 01/Nov/17

$$thegrounddidnotjustcatchthe$$$$ball,iteventhrewitbackwith$$$$speed{ev}_0.$$$$\Rightarrow changeinmomentum$$$$broughtabout=m[{ev}_0-(-v_0)].$$

Commented by Tinkutara last updated on 01/Nov/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

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