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Question Number 112971 by ajfour last updated on 10/Sep/20

Commented by ajfour last updated on 11/Sep/20

$A s m a l l s p h e r e o f r a d i u s r, m a s s m,$ $c o l l i d e s e l a s t i c a l l y w i t h a n o t h e r$ $s p h e r e o f s a m e d e n s i t y a s t h a t o f$ $s m a l l e r s p h e r e . F i n d r a d i u s o f$ $l a r g e r s p h e r e s u c h t h a t i t s r a n g e$ $x b e m a x i m u m .$
	Answered by mr W last updated on 12/Sep/20

	Commented by mr W last updated on 12/Sep/20

	$M = {(\frac{R}{r})}^{3} m$ $\sin θ = \frac{R - r}{R + r} = 1 - \frac{2}{\frac{R}{r} + 1}$ $\Rightarrow \frac{R}{r} = \frac{2}{1 - \sin θ} - 1$ $m u = M V \cos θ + m v$ $\Rightarrow u = {(\frac{R}{r})}^{3} V \cos θ + v$ $\frac{{m u}^{2}}{2} = \frac{{M V}^{2}}{2} + \frac{{m v}^{2}}{2}$ $\Rightarrow u^{2} = {(\frac{R}{r})}^{3} V^{2} + v^{2}$ $\Rightarrow u^{2} = {(\frac{R}{r})}^{3} V^{2} + {[u - {(\frac{R}{r})}^{3} \cos θ V]}^{2}$ $\Rightarrow [1 + {(\frac{R}{r})}^{3} \cos^{2} θ] V = 2 u \cos θ$ $\Rightarrow V = \frac{2 u \cos θ}{1 + {(\frac{R}{r})}^{3} \cos^{2} θ}$ $0 = V \sin θ \times \frac{x}{V \cos θ} - \frac{g}{2} {(\frac{x}{V \cos θ})}^{2}$ $\Rightarrow g x = \sin 2 θ V^{2}$ $\Rightarrow g x = \sin 2 θ {[\frac{2 u \cos θ}{1 + {(\frac{R}{r})}^{3} \cos^{2} θ}]}^{2}$ $\Rightarrow \frac{g x}{8 u^{2}} = \frac{\sin θ \cos^{3} θ}{{[1 + {(\frac{R}{r})}^{3} \cos^{2} θ]}^{2}}$ $\Rightarrow \frac{g x}{8 u^{2}} = \frac{\sin θ \cos^{3} θ}{{[1 + {(\frac{2}{1 - \sin θ} - 1)}^{3} \cos^{2} θ]}^{2}}$ $x_{m a x} i s a t θ = 6.9827 °$ $o r \frac{R}{r} = 1.277$
	Commented by mr W last updated on 13/Sep/20

	$t o b e h o n e s t, i^{'} m n o t s u r e i f {i t}^{'} s$ $c o r r e c t t o t a k e t h e d i r e c t i o n o f V$ $a t a n g l e θ .$
	Commented by ajfour last updated on 14/Sep/20

	$T h a n k y o u S i r, P e r f e c t s o l u t i o n!$ $b u t i f w e l e t \frac{R}{r} = \frac{2}{1 - \sin θ} - 1 = λ$ $\frac{g x}{8 u^{2}} = \frac{8 λ \sqrt{λ} (λ - 1)}{{(4 λ^{4} + λ^{2} + 2 λ + 1)}^{2}}$ $I t s m a x o c c u r s w h e n$ $44 λ^{5} - 52 λ^{4} + 3 λ^{3} - 7 λ^{2} - 7 λ + 3 = 0$ $w e g e t t h r e e r o o t s, o n l y o n e o f t h e m$ $i s > 1, s o t h a t λ = \frac{R}{r} \approx 1.277$
	Commented by mr W last updated on 14/Sep/20

	$g r e a t!$
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