Question

A bullet loses $\dfrac{1}{{20}}$ of its initial velocity after penetrating a plank. How many planks are required to stop the bullet?
A. 6
B. 9
C. 11
D. 13

Answer 1

Hint: We can solve this problem using the third equation of motion
${v^2} = {u^2} + 2aS$
Where, $v$ is the final velocity, $u$ is initial velocity, a is acceleration and S is the distance travelled. Suppose the number of planks required to stop the bullet is n. Then, by finding the equations of motion for this n number of planks and the equation for one plank and solving these two equations we can find the value of the number of planks required.

Complete answer:
It is given that a bullet loses $\dfrac{1}{{20}}$ of its initial velocity after penetrating through a plank.
 We need to find the number of planks which are required to stop the bullet.
Suppose that plank has a thickness d.
Let us use the third equation of motion.
The third equation of motion is given as,
$ \Rightarrow $${v^2} = {u^2} + 2aS$ -------------(1)
Where, v is the final velocity, u is initial velocity, a is acceleration and S is the distance travelled.
We need to find the number of planks when the bullet finally stops.
So, the final velocity is zero.
$ \Rightarrow $$V = 0$
When there are $n$ number of planks then the total distance will be n times the thickness of one plank.
$ \Rightarrow $So, $S = n \times d$
On substituting these values in equation 1 we get
$ \Rightarrow $$0 = {u^2} + 2a \times n \times d$.
$ \Rightarrow n = - \dfrac{{{u^2}}}{{2ad}}$ -------------(2)
Velocity of the bullet when it passes through each plank gets reduced by $\dfrac{1}{{20}}$of its initial value.
So, after crossing 1 plank the new velocity will be
$ \Rightarrow $$v' = u - \dfrac{1}{{20}}u$
$ \Rightarrow v' = \dfrac{{19u}}{{20}}$
If you write the equation of motion when it crosses one plank, we will get
$ \Rightarrow $${v'^2} = {u^2} + 2a \times d$
$ \Rightarrow {v'^2} - {u^2} = 2ad$
On substituting the value of $v'$, we get
$ \Rightarrow {\left( {\dfrac{{19u}}{{20}}} \right)^2} - {u^2} = 2ad$
$ \Rightarrow \dfrac{{361}}{{400}}{u^2} - {u^2} = 2ad$
$ \Rightarrow \dfrac{{ - 39}}{{400}}{u^2} = 2ad$ ………………………...(3)
On substituting the value of $2ad$ we got in equation 3 in equation 2 we get
 $ \Rightarrow n = \dfrac{{ - {u^2}}}{{\dfrac{{ - 39}}{{400}}{u^2}}}$
$\therefore n = \dfrac{{400}}{{39}} = 10.25$
Since this value is greater than 10 but we require minimum 11 planks.
So, the number of planks required to stop the bullet is 11.

Hence correct answer is option C.

Note:
When velocity is decreasing uniformly, we get a constant acceleration. The equation of motion should be used only in cases where there is constant acceleration. If the acceleration is varying continuously then the motion will be more complex that we will not be able to solve it using the equations of motion.