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Question Number 54289 by ajfour last updated on 01/Feb/19

Commented by ajfour last updated on 01/Feb/19

$F i n d u s u c h t h a t u p p e r b l o c k t o p p l e s,$ $i f p a r t i c l e o f m a s s m h i t s e l a s t i c a l l y .$ $A l s o f i n d w h e r e t h e p a r t i c l e$ $h i t s t h e g r o u n d .$
Commented by tanmay.chaudhury50@gmail.com last updated on 01/Feb/19

$f r o m y o u t u b e i s a w t h i s v i d e o . . . e x c e l l e n t e x p l a n a t i o n$ $h e n c e a t t a c h e d . . .$
Commented by tanmay.chaudhury50@gmail.com last updated on 01/Feb/19

Commented by mr W last updated on 01/Feb/19

$u ⩾ 5 \sqrt{\frac{3 μ g b}{2}} ?$
Commented by ajfour last updated on 01/Feb/19

$g i v e n u = \frac{5}{2} \sqrt{μ g b}; b u t j u s t n o w i$ $g o t t h e s a m e r e s u l t a s y o u r s S i r .$
	Answered by mr W last updated on 02/Feb/19

	$s u c h t h a t t h e u p p e r b l o c k t o p p l e s,$ $t h e l o w e r b l o c k m u s t m o v e a t l e a s t$ $a d i s t a n c e o f s = 2 b a f t e r t h e h i t .$ $i n t h i s d i s t a n c e t h e f r i c t i o n f o r c e$ $i s f = μ (2 m + 4 m) g = 6 μ m g a n d t h e$ $e n e r g y l o s t d u e t o f r i c t i o n i s f s,$ $i . e . 6 μ m g \times 2 b = 12 μ m g b . t h a t i t t o s a y$ $t h e l o w e r b l o c k m u s t h a v e a k i n e t i c$ $e n e r g y o f a t l e a s t 12 μ m g b a f t e r t h e h i t .$ $K E = \frac{1}{2} (4 m) v^{2} ⩾ 12 μ m g b$ $\Rightarrow v = \sqrt{6 μ g b} = v e l o c i t y o f l o w e r b l o c k a f t e r h i t$ $l e t u_{1} = v e l o c i t y o f p a r t i c l e a f t e r h i t (\leftarrow)$ $u = u_{1} + v$ $s i n c e m u = - {m u}_{1} + 4 m v,$ $\Rightarrow u = - u_{1} + 4 v$ $\Rightarrow u = - (u - v) + 4 v$ $\Rightarrow 2 u = 5 v$ $\Rightarrow u = \frac{5 v}{2} ⩾ \frac{5}{2} \sqrt{6 μ g b} = 5 \sqrt{\frac{3 μ g b}{2}}$ $u_{1} = u - v = \frac{5 v}{2} - v = \frac{3 v}{2} = 3 \sqrt{\frac{3 μ g b}{2}}$ $p o s i t i o n w h e r e p a r t i c l e h i t s t h e g r o u n d :$ $d = u_{1} \sqrt{\frac{2 b}{g}} = 3 \sqrt{3 μ} b$
	Commented by ajfour last updated on 01/Feb/19

	$i h a v e g o t t o k n o w$ $d = 6 b \sqrt{3 μ} .$
	Commented by mr W last updated on 01/Feb/19

	$p l e a s e c h e c k s i r!$ $h = b = \frac{{g t}^{2}}{2}$ $d = u_{1} t = u_{1} \sqrt{\frac{2 b}{g}} = 3 \sqrt{\frac{3 μ g b}{2} \times \frac{2 b}{g}} = 3 \sqrt{3 μ} b$
	Commented by ajfour last updated on 01/Feb/19

	$s o r r y s i r, i t w a s a s k e d i f v e l o c i t y$ $o f p a r t i c l e w e r e d o u b l e o f p r e v i o u s l y$ $f o u n d m i n . v e l o c i t y, w h e r e w o u l d i t$ $h i t t h e g r o u n d ?$
	Commented by ajfour last updated on 01/Feb/19

	$A s p e r m y p o s t e d q u e s t i o n, a l l y o u r$ $r e s u l t s a n d e x p l a n a t i o n, i b e l i e v e$ $a r e c o r r e c t, S i r . T h a n k s a g a i n!$
	Answered by ajfour last updated on 01/Feb/19

	$m u = 4 m V - m v$ $u = V + v$ $\Rightarrow u = 4 V + V - u$ $\Rightarrow V = \frac{2 u}{5}$ $\frac{1}{2} (4 m) V^{2} ⩾ μ (6 m g) (2 b)$ $\Rightarrow 2 {(\frac{2 u}{5})}^{2} ⩾ 12 μ g b$ $\Rightarrow u ⩾ 5 \sqrt{\frac{3 μ g b}{2}} .$
	Commented by mr W last updated on 01/Feb/19

	$t h a t i s t h e c o r r e c t r e s u l t!$
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