A-ball-is-bouncing-down-a-flight-of-stairs-The-coefficiate-of-restitution-is-e-The-height-of-each-step-d-and-the-ball-descends-one-step-at-each-bounce-After-each-bounce-it-rebounds-to-heigt-h-above

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Question Number 51107 by peter frank last updated on 24/Dec/18

$$Aballisbouncingdown$$$$aflightofstairs.The$$$$coefficiateofrestitution$$$$ise.Theheightofeachstep$$$$dandtheballdescends$$$$onestepateachbounce.$$$$Aftereachbounceitrebounds$$$$toheigthabovethenext$$$$lowerstep.Theheighthis/$$$$largeenoughcomparewith$$$$widthofastepthat$$$$theempactsareeffectively$$$$headon.showthat$$$$h=\frac{d}{1-e^2}$$$$$$

Commented by peter frank last updated on 24/Dec/18

$$sirwearegiventhattheballdescend$$$$onestepateachbounce.so$$$$ballfallsadistancehfrom$$$$itsheighestpositionand$$$$reboundstoadistance$$$$(h-d)ithink.$$

Answered by mr W last updated on 24/Dec/18

$$heightoftheballbeforerebound:h$$$$heightoftheballafterrebound:h-d$$$$velocityoftheballbeforerebound:v_{before}$$$$velocityoftheballafterrebound:v_{after}$$$$E_{before}=mgh=\frac{1}{2}{mv}_{before}^2$$$$E_{after}=mg(h-d)=\frac{1}{2}{mv}_{after}^2$$$$\Rightarrow \frac{h-d}{h}={\left(\frac{v_{after}}{v_{before}}\right)}^2$$$$since{v}_{after}={ev}_{before}$$$$\Rightarrow \frac{h-d}{h}=e^2$$$$\Rightarrow h-d=e^{2}h$$$$\Rightarrow h=\frac{d}{1-e^2}$$

Answered by peter frank last updated on 24/Dec/18

$$e=\frac{velocityofseparation}{velocityofapproach}$$$$e=\frac{\sqrt{2g(h-d)}}{\sqrt{2gh}}$$$$e\sqrt{2gh}=\sqrt{2g(h-d)}$$$$e^2.2gh=2g(h-d)$$$$e^2.2gh=2gh-2gd$$$$e^2=1-\frac{d}{h}$$$$1-e^2=\frac{d}{h}$$$$h=\frac{d}{1-e^2}$$$$henceshown$$$$$$

Commented by mr W last updated on 24/Dec/18

$$sorry,imisreadtheinformationabouth.$$$$now{it}^{\prime }sfixed.$$

Commented by peter frank last updated on 24/Dec/18

$$okaysir.$$

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