Question

When a bullet is fired at a target, its velocity decreases by half after penetrating $30cm$ into it. The additional thickness it will penetrate before coming to rest is

• A. $30cm$
• B. $40cm$
• C. $10cm$
• D. $50cm$

Answer 1

The correct option is C

$10cm$

Step 1. Given data:

Penetrating depth = $30cm$

Let the velocity of the bullet when it strikes the target is $vcm{s}^{-1}$

Then Final velocity = $\frac{1}{2}$($v$).

Step 2. Formula used:

Considering the equation of motion, $v^2=u^2+2as$

Where $v$=Final velocity, $u$= initial velocity, $a$= acceleration, $s$= distance,

Step 3. Calculating acceleration:

Putting the values in the above equation, we get:

$$\begin{gathered} {\left(\frac{v}{2}\right)}^2=v^2+2.a.\left(30\right) \\ 60a=\frac{v^2}{4}-v^2=\frac{v^2-4v^2}{4}=\frac{-3v^2}{4} \\ a=\frac{-v^2}{80}cm{s}^{-2} \end{gathered}$$

Step 4: Calculating additional thickness:

Let the bullet further penetrates $xcm$ before coming to rest, therefore

Considering the equation of motion, $v^2=u^2+2as$ again.

Then $s=x$, Final velocity=0, initial velocity =$\frac{v}{2}$

Then $$\begin{gathered} 0=\frac{v^2}{4}+2\left(\frac{-v^2}{80}\right).x \\ \frac{v^2.x}{40}=\frac{v^2}{4} \\ orx=10cm \end{gathered}$$

Thus, the correct option is option C.