Question 67 of 308

A bullet loses ${\left(\dfrac{1}{n}\right)}^{th}$ of its velocity passing through one plank. Considering uniform retardation, the number of such planks that are required to stop the bullet can be:

AInfiniteB$n$C$\dfrac{{n}^{2}}{\left(2n-1\right)}$D$\dfrac{2{n}^{2}}{\left(n-1\right)}$

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Answer: (C) $\dfrac{{n}^{2}}{\left(2n-1\right)}$

Let $t$ be the thickness of the plank
${\left\{{υ}_{0}\left(1-\dfrac{1}{n}\right)\right\}}^{2}={υ}_{0}^{2}-2at$
$∴2at={υ}_{0}^{2}-{υ}_{0}^{2}{\left(1-\dfrac{1}{n}\right)}^{2}$
If $m$ number of planks are required to stop a bullet, then
$0={υ}_{0}^{2}-\left(2at\right)m$
$∴m=\dfrac{{υ}_{0}^{2}}{2at}$
$=\dfrac{{υ}_{0}^{2}}{{υ}_{0}^{2}\left\{1-{\left(1-\dfrac{1}{n}\right)}^{2}\right\}}$
$=\dfrac{{n}^{2}}{{n}^{2}-{\left(n-1\right)}^{2}}=\dfrac{{n}^{2}}{\left(2n-1\right)}$