The engine of a motorcycle can produce a maximum acceleration of $5 m s^{- 2}$ . Its brakes can produce a maximum retardation $10 m s^{- 2}$ . The minimum time (in sec) in which it can cover a distance of 1.5 km is ____________

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answer is 30.

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Detailed Solution

Let V be the final velocity acquired in this case when accelerated for time $t_{1}$ . Let $t_{2}$ be the time for which it retards. Let $S_{1}$ and $S_{2}$ be the distances covered in these two cases then
$V^{2} = 2 a_{1} S_{1} = 2 a_{2} S_{2}$
$\Rightarrow S_{1} = \frac{V^{2}}{2 a_{1}} = \frac{V^{2}}{10} and S_{2} = \frac{V^{2}}{2 a_{2}} = \frac{V^{2}}{20}, H e r e, S_{1} + S_{2} = 1500 m$

$\Rightarrow \frac{V^{2}}{10} + \frac{V^{2}}{20} = 1500 \Rightarrow V = 100 m s^{- 1}$
Similarly $V = a_{1} t_{1} = a_{2} t_{2}$

$\Rightarrow 100 = 5 t_{1} = 10 t_{2} \Rightarrow t_{1} = 20 s {and t}_{2} = 10 s \Rightarrow t_{1} + t_{2} = 30 s$