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 the next lower step.The height h is/ large enough compare with width of a step that the empacts are effectively head on.show that h=(d/(1−e^2 ))


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

$A b a l l i s b o u n c i n g d o w n$ $a f l i g h t o f s t a i r s . T h e$ $c o e f f i c i a t e o f r e s t i t u t i o n$ $i s e . T h e h e i g h t o f e a c h s t e p$ $d a n d t h e b a l l d e s c e n d s$ $o n e s t e p a t e a c h b o u n c e .$ $A f t e r e a c h b o u n c e i t r e b o u n d s$ $t o h e i g t h a b o v e t h e n e x t$ $l o w e r s t e p . T h e h e i g h t h i s /$ $l a r g e e n o u g h c o m p a r e w i t h$ $w i d t h o f a s t e p t h a t$ $t h e e m p a c t s a r e e f f e c t i v e l y$ $h e a d o n . s h o w t h a t$ $h = \frac{d}{1 - e^{2}}$
Commented by peter frank last updated on 24/Dec/18

$s i r w e a r e g i v e n t h a t t h e b a l l d e s c e n d$ $o n e s t e p a t e a c h b o u n c e . s o$ $b a l l f a l l s a d i s t a n c e h f r o m$ $i t s h e i g h e s t p o s i t i o n a n d$ $r e b o u n d s t o a d i s t a n c e$ $(h - d) i t h i n k .$
	Answered by mr W last updated on 24/Dec/18

	$h e i g h t o f t h e b a l l b e f o r e r e b o u n d : h$ $h e i g h t o f t h e b a l l a f t e r r e b o u n d : h - d$ $v e l o c i t y o f t h e b a l l b e f o r e r e b o u n d : v_{b e f o r e}$ $v e l o c i t y o f t h e b a l l a f t e r r e b o u n d : v_{a f t e r}$ $E_{b e f o r e} = m g h = \frac{1}{2} {m v}_{b e f o r e}^{2}$ $E_{a f t e r} = m g (h - d) = \frac{1}{2} {m v}_{a f t e r}^{2}$ $\Rightarrow \frac{h - d}{h} = {(\frac{v_{a f t e r}}{v_{b e f o r e}})}^{2}$ $s i n c e v_{a f t e r} = {e v}_{b e f o r e}$ $\Rightarrow \frac{h - d}{h} = e^{2}$ $\Rightarrow h - d = e^{2} h$ $\Rightarrow h = \frac{d}{1 - e^{2}}$
	Answered by peter frank last updated on 24/Dec/18

	$e = \frac{v e l o c i t y o f s e p a r a t i o n}{v e l o c i t y o f a p p r o a c h}$ $e = \frac{\sqrt{2 g (h - d)}}{\sqrt{2 g h}}$ $e \sqrt{2 g h} = \sqrt{2 g (h - d)}$ $e^{2} . 2 g h = 2 g (h - d)$ $e^{2} . 2 g h = 2 g h - 2 g d$ $e^{2} = 1 - \frac{d}{h}$ $1 - e^{2} = \frac{d}{h}$ $h = \frac{d}{1 - e^{2}}$ $h e n c e s h o w n$
	Commented by mr W last updated on 24/Dec/18

	$s o r r y, i m i s r e a d t h e i n f o r m a t i o n a b o u t h .$ $n o w {i t}^{'} s f i x e d .$
	Commented by peter frank last updated on 24/Dec/18

	$o k a y s i r .$
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