A particle moves along a straight line such that its velocity, v m s^(−1) , is given by v=t^3 −4t^2 +3t, where t is time, in seconds, after passing through fixed point O. Find the total distance, in m, travelled by the particle until the particle returned to the fixed point O for the second time.


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Question Number 114145 by ZiYangLee last updated on 17/Sep/20

$A particle moves along a straight line$ $such that its velocity, v m s^{- 1}, is given$ $by v = t^{3} - 4 t^{2} + 3 t, where t is time, in$ $seconds, after passing through fixed$ $point O .$ $Find the total distance, in m, travelled$ $by the particle until the particle returned$ $to the fixed point O for the second time .$
	Answered by mr W last updated on 17/Sep/20

	$s (t) = \int_{0}^{t} v d t = \frac{t^{4}}{4} - \frac{4 t^{3}}{3} + \frac{3 t^{2}}{2}$ $s (t) = \frac{t^{4}}{4} - \frac{4 t^{3}}{3} + \frac{3 t^{2}}{2}$ $s_{m a x} i s w h e n v = 0,$ $t^{3} - 4 t^{2} + 3 t = 0$ $(t^{2} - 4 t + 3) t = 0$ $(t - 1) (t - 3) t = 0$ $t = 0$ $t = 1$ $t = 3$ $a t t = 1 : s_{m a x} = \frac{1^{4}}{4} - \frac{4 \times 1^{3}}{3} + \frac{3 \times 1^{2}}{2} = \frac{5}{12}$ $t o t a l d i s t a n c e t r a v e l l e d :$ $s = 2 \times s_{m a x} = \frac{5}{6} m$
	Commented by ZiYangLee last updated on 17/Sep/20

	$oops sorry the answer is 5 . 335 m instead$
	Commented by mr W last updated on 17/Sep/20

	${i t}^{'} s u p o n w h a t i s m e a n t w i t h ‘ ‘ r e t u r n$ $t o p o i n t O f o r t h e s e c o n d {t i m e}^{″} .$ $w h a t i m e a n t i s t h a t t h e o b j e c t l e a v e s$ $p o i n t O a n d c o m e s b a c k . b u t i f i t$ $m e a n s t h a t t h e o b j e c t l e a v e s p o i n t O$ $a n d c o m e s b a c k a n d l e a v e s a g a i n a n d$ $c o m e s b a c k a g a i n, t h e n t h e r e s u l t i s :$ $a t t = 3 : s_{m i n} = \frac{3^{4}}{4} - \frac{4 \times 3^{3}}{3} + \frac{3 \times 3^{2}}{2} = - \frac{9}{4}$ $\Rightarrow t o t a l d i s t a n c e m o v e d :$ $2 \times \frac{5}{12} + 2 \times \frac{9}{4} = \frac{16}{3} = 5.333 m$
	Commented by mr W last updated on 17/Sep/20

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