Two cars are travelling on a straight road. The car c1 is going at a constant speed of 9m/s. 40 metres ahead of c1, car c2 initially at rest, but starts moving at an acceleration of 1m/s^2.During their motion, the car c1 overtakes c2, but soon c2 overtakes c1. Determine the maximum lead that c1 can have from c2:
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a) 0.5m   b)9.5m   c)81m  d)90m

Dear Student,

Please find below the solution to the asked query:

If the separation between the C₁ and C₂ is maximum, then the difference of the first car displacement and the displacement of the second car displacement should have its maximum value. Therefore,

 $$\begin{gathered} S_1=9t \\ {S}_2=\frac{1}{2}\times 1\times t^2 \\ So, the difference is, \\ ∆S=S_1-S_2=9t-\frac{t^2}{2} \\ For this to be maximum, \\ \frac{d\left(∆S\right)}{dt}=0\Rightarrow \frac{d}{dt}\left\{9t-\frac{t^2}{2}\right\}=0\Rightarrow 9-t=0\Rightarrow t=9 sec. \\ So, \\ ∆S=9\times 9-\frac{9^2}{2}=40.5 m \\ Therefore, initially the first car is 40 m behind the second car. \\ Therefore the maximum lead is 0.5 m \end{gathered}$$

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