Question

Percentage of loss of velocity after passing through a plank is $25\% $ . Number of planks required to bring bullet to rest is
A. $4$
B. $5$
C. $2$
D. $3$

Answer 1

Hint:This problem is based on the inelastic collision. In an inelastic collision the bullet will be embedded through the plank and its velocity decreases and finally the velocity will become zero, that is, the bullet will come to rest. We need to find the number of planks that the bullet passes before coming to rest.

Formula used:
${v^2} - {u^2} = 2as$
Where, $u$ = initial velocity, $v$ = final velocity, $a$ = acceleration and $s$ = displacement.

Complete step by step answer:
Let the initial velocity of the body which is passing through be $x$ and final velocity will be $\dfrac{3}{4}x$
That is, $u = x$ ……… $\left( 1 \right)$
and , $v = \left( {100 - 25} \right)\% $ of $x$
Therefore, $v = \dfrac{{75 \times x}}{{100}}$
On simplifying the above equation, we get final velocity as
$v = \dfrac{{3 \times x}}{4}$………. $\left( 2 \right)$
We know that, from the equation of linear motion
${v^2} - {u^2} = 2as$ ……….. $\left( 3 \right)$
Substituting equation $\left( 1 \right)$ and equation$\left( 2 \right)$ in equation $\left( 3 \right)$ , we get
${\left( {\dfrac{{3x}}{4}} \right)^2} - {x^2} = - 2as$ ($a = - a$ , because the body will be decelerate when it passes through plank)

On simplifying the above equation, we get
$\left( {\dfrac{{9{x^2}}}{{16}}} \right) - {x^2} = - 2as$
On cross multiplying the above equation becomes,
$\dfrac{{9{x^2} - 16{x^2}}}{{16}} = - 2as$
On simplifying the above equation,
$\dfrac{{ - 7{x^2}}}{{16}} = - 2as$
Therefore, $as = \dfrac{{7{x^2}}}{{32}}$ ………. $\left( 4 \right)$
Since the body will come to rest therefore the velocity will $0$
Hence equation $\left( 3 \right)$ becomes
$0 = {x^2} - 2as \times n$ ……… $\left( 5 \right)$
Where, $n = $ Number of planks.
Substituting equation $\left( 4 \right)$ in equation$\left( 5 \right)$, we get
$0 = {x^2} - 2\left( {\dfrac{{7{x^2}}}{{32}}} \right) \times n$ ………. $\left( 6 \right)$
On simplifying equation $\left( 6 \right)$ , we get
$\therefore n = 2.2$
Therefore number planks will be $3$ .

Hence, the correct option is D.

Note: It should be noted that the final velocity will be $75\% $ of the initial velocity because it is given that the percentage loss of velocity passing through a plank is $25\% $.And also the value of $''a''$ should be considered as negative because the velocity reduces as the bullet passes through the plank, that is deceleration will take place.