Question

A bullet leaving the muzzle of a rifle barrel with a velocity $v$ penetrates a plank and loses ${\frac{1}{5}}^{th}$ of its velocity. It then strikes the second plank, which it just penetrates through. Find out the ratio of the thickness of the planks. [Supposing the average resistance to the penetration is the same in both the case]

• A. $\frac{3}{4}$
• B. $\frac{9}{16}$
• C. $\frac{16}{25}$
• D. $\frac{9}{25}$

Answer 1

The correct option is B $\frac{9}{16}$

Let $F_R$ be the resistive force for each plank,
$v$ be velocity of bullet before it hits the first plank

As bullet looses ${\frac{1}{5}}^{th}$ of its velocity.
Velocity of bullet when it leaves the first plank is $v-\frac{v}{5}=\frac{4v}{5}$

Let $t_1,t_2$ be thickness of first and second plank.

From Work energy theorem-
Work done by resistance force $=\Delta KE$

$F_R.t_1=\frac{1}{2}m{\left(\frac{4v}{5}\right)}^2-\frac{1}{2}m{v}^2$
$F_R.t_1=-\frac{1}{2}m\left(\frac{9}{25}\right)v^2\dots \dots \left(1\right)$
For second plank:
$F_R.t_2=0-\frac{1}{2}m{\left(\frac{4v}{5}\right)}^2$
$F_R.t_2=-\frac{1}{2}m\left(\frac{16}{25}\right)v^2\dots \dots \left(2\right)$
Dividing (1) by (2) we get
$\frac{t_1}{t_2}=\frac{9}{16}$