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Question Number 170120 by mr W last updated on 16/May/22

Commented by mr W last updated on 17/May/22

$A t r u c k w i t h m a s s M i s m o v i n g o n$ $a h o r i z o n t a l a n d s t r a i g h t r o a d w i t h$ $s p e e d v . A s m a l l b o x w i t h m a s s m l i e s$ $a t t h e e n d o f t h e l u g g a g e c o m p a r t m e n t$ $o f l e n g t h L . T h e f r i c t i o n c o e f f i c i e n t$ $b e t w e e n t i r e s o f t r u c k a n d t h e r o a d$ $i s μ, a n d t h a t b e t w e e n t h e b o x a n d t h e$ $t r u c k i s μ / 2 . T h e t r u c k o b t a i n s a f u l l$ $b r a k e a n d i t s t o p s e x a c t l y a t t h e$ $i n s t a n t a s t h e b o x r e a c h e s t h e o t h e r$ $e n d o f t h e l u g g a g e c o m p a r t m e n t .$ $F i n d t h e s p e e d o f t h e t r u c k a s i t s t a r t s$ $t o b r a k e .$
Commented by ajfour last updated on 18/May/22

$i h a d a g a i n j u s t n o w s i r, b u t$ $g o t d e l e t e d, s h a l l a n s w e r s o o n$ $a g a i n . .$
	Answered by mr W last updated on 19/May/22

	$N_{T} = (M + m) g$ $f_{T} = μ N_{T} = μ (M + m) g$ $N_{B} = m g$ $f_{B} = \frac{μ N_{B}}{2} = \frac{μ m g}{2}$ ${m a}_{B} = f_{B} = \frac{μ m g}{2}$ $\Rightarrow a_{B} = \frac{μ g}{2}$ ${M a}_{T} = - f_{B} + f_{T} = - \frac{μ m g}{2} + μ (M + m) g$ $\Rightarrow a_{T} = μ (1 + \frac{m}{2 M}) g$ $s a y t h e t r u c k t r a v e l e d a d i s t a n c e d$ $i n t i m e t, t h e n t h e b o x t r a v e l e d$ $a d i s t a n c e o f d + L i n t h i s t i m e .$ $t = \frac{v}{a_{T}}$ $d = \frac{v^{2}}{2 a_{T}}$ $d + L = v t - \frac{a_{B} t^{2}}{2}$ $\frac{v^{2}}{2 a_{T}} + L = \frac{v^{2}}{a_{T}} - \frac{a_{B}}{2} {(\frac{v}{a_{T}})}^{2}$ $2 {L a}_{T} = (1 - \frac{a_{B}}{a_{T}}) v^{2}$ $2 L μ (1 + \frac{m}{2 M}) g = (1 - \frac{\frac{μ g}{2}}{μ (1 + \frac{m}{2 M}) g}) v^{2}$ $μ g L (2 + \frac{m}{M}) = (\frac{1 + \frac{m}{M}}{2 + \frac{m}{M}}) v^{2}$ $l e t ξ = \frac{m}{M}$ $μ g L (2 + ξ) = (\frac{1 + ξ}{2 + ξ}) v^{2}$ $\Rightarrow v = (2 + ξ) \sqrt{\frac{μ g L}{1 + ξ}}$
	Commented by ajfour last updated on 17/May/22

	$d o i n e e d t o e x p l a i n m y w a y,$ $i n m o r e d e t a i l s i r ?$ $i t s s h o r t e r . .$
	Commented by mr W last updated on 17/May/22

	$p l e a s e s i r!$
	Commented by mr W last updated on 18/May/22

	Commented by Tawa11 last updated on 08/Oct/22

	$Great sir$
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