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  4. A bullet when fired at a target with a velocity of 100 m/s penetrates one metre into it. If the bullet is fired at a similar target with a thickness 0.5 m . Then, it will emerge from it with a velocity of

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Q. A bullet when fired at a target with a velocity of $ 100 \,m/s $ penetrates one metre into it. If the bullet is fired at a similar target with a thickness $ 0.5 \,m $ . Then, it will emerge from it with a velocity of

UPSEEUPSEE 2012

Solution:

Let $v$ be the velocity with which the bullet will energy. Now,
Change in kinetic energy $=$ Work done.
In first case,
$\frac{1}{2} \,m\,(100)^{2}=\frac{1}{2} \,m\,(0)^{2}\,=\,F \times 1\,\,\,\,\,\,\, ...(i)$
In second case,
$\frac{1}{2}\, m\,(100)^{2}-\frac{1}{2} \,m v^{2}\,=\,F \times 0.5\,\,\,\,\,\,\, ...(ii)$
Dividing Eq. (ii) by Eq. (i), we get
$\frac{\left(100^{2}\right)-v^{2}}{(100)^{2}} =\frac{0.5}{1}=\frac{1}{2} $
$1-\frac{v^{2}}{(100)^{2}} =\frac{1}{2} $
$\left(\frac{v}{100}\right)^{2} =\frac{1}{2} $
$v= \frac{100}{\sqrt{2}}=50 \sqrt{2} \,m / s$