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Q.
When a bullet is fired at a target, its velocity decreases by half after penetrating 30 cm into it. The additional thickness it will penetrate before coming to rest is
Let the velocity of the bullet when it strikes the target is $v cms^{-1}$
After penetrating 30 cm, velocity becomes half ie.$\frac{v}{2}$
From equation $v^2 = u^2 + 2as$
hance $\left(\frac{v}{2}\right)^2=v^2+2a \times 30$
or $-60a=v^2-\frac{v^2}{4}$
or $-60a = \frac{3v^2}{4}$
$a=-\frac{v^2}{80}cms^{-2}$
Let the bullet further penetrates x cm before coming to rest, therefore
$v^{'2}=u^{'2}+2as'$
$0=\left(\frac{v}{2}\right)^2+2\left(-\frac{v^2}{80}\right)x$
$\frac{v^2x}{40}=\frac{v^2}{4}$
x = 10 cm