Question

A rifle shoots a bullet with a muzzle velocity of $400\text{m/s}$at a small target $400\text{m}$ away. The height above the target at which the bullet must be hit to aim the target is $(g=10\text{m}{{\text{s}}^{-1}})$.
$\begin{aligned} & \text{A)1m} \\\ & \text{B)5m} \\\ & \text{C)10m} \\\ & \text{D)0}\text{.5m} \\\ \end{aligned}$

Answer 1

Hint: When a bullet is fired, it acts like a projectile in the air. Due to gravity of earth, it will deflect downwards as it travels. So, it must be initially fired at a certain height above the target to make it actually reach the target. This height can be calculated using projectile equations of motion.

Complete step-by-step answer:
Let the velocity of the bullet along the x-axis is ${{v}_{x}}$and along the y-axis is ${{v}_{y}}$. Then we can write,
${{v}_{x}}=400\cos \theta \text{ and }{{v}_{y}}=400\sin \theta \text{ }$( as in the question the muzzle velocity is given as $400\text{m/s}$)
As $\theta$ is very small, so
${{v}_{x}}=400\text{m/s and }{{v}_{y}}=0$
Let the time taken by the target to hit the target is $t\text{ seconds}$. Then, we have,
$t=\dfrac{{{v}_{x}}}{d}$
where, $d$ is the distance between the rifle and the target. Therefore,
$\begin{aligned} & t=\dfrac{400\text{m}}{400\text{m/s}} \\\ & \Rightarrow t=1\text{ s} \\\ \end{aligned}$
Now, if the height covered by the bullet to hit the target is $y$, then
$\begin{aligned} & y={{v}_{y}}+\dfrac{1}{2}g{{t}^{2}} \\\ & \Rightarrow y=\dfrac{1}{2}g{{t}^{2}}\text{ (since, }{{v}_{y}}=0) \\\ & \Rightarrow y=\dfrac{1}{2}\times 10\times {{(1)}^{2}} \\\ & \Rightarrow y=5\text{m} \\\ \end{aligned}$
In the above calculations $g$ is the acceleration due to gravity and it is already mentioned in the question that $g=10\text{m}{{\text{s}}^{-1}}$.
Therefore the answer is option $\text{B)5m}$.

Note: In this question, it is given that the target is small and that is the reason why we can consider the angle made by the bullet while being fired with the horizontal to be small, if the target would not have been mentioned to be small, then it would not have been proper to consider the angel to be small. In that case the calculations would have been different.