Question

Two guns A and B can fire bullets at speeds 1 km/s and 2 km/s respectively. From a point on a horizontal ground, they are fired in all possible directions. The ratio of maximum areas covered by the bullets fired by the two guns, on the ground is
A. 1:2
B. 1:4
C. 1:8
D. 1:16

Answer 1

Here the bullets undergo projectile motion, i.e. the motion of an object thrown or projected into the air. Its path is named as trajectory. The range is the maximum horizontal distance that it undergoes. When we complete a circle with R as the radius, we can obtain the maximum area covered by the bullets.

Formula used: Formula for range $R=\dfrac{{{u}^{2}}\sin 2\theta }{g}$, where $u$is the initial velocity , $g$is the acceleration due to gravity and $\theta$ is the initial launch angle.

Complete step by step answer:
We’re given that the initial velocity of the bullets from the first gun is $u1=1m/s$
And the initial velocity of the bullets from the second gun is $u2=2m/s$
We know that the range $R=\dfrac{{{u}^{2}}\sin 2\theta }{g}$---------(1)
and area of a circle is $A=\pi {{R}^{2}}$---------(2)
Here, we take Range as the radius.
From (1) and (2) we can infer that
$A\propto {{R}^{2}}$
$\Rightarrow A\propto {{u}^{4}}$
$\therefore$ $\dfrac{A1}{A2}=\dfrac{u{{1}^{4}}}{u{{2}^{4}}}$ $={{(\dfrac{1}{2})}^{4}}=\dfrac{1}{16}$
Hence the ratio is 1:16

So, the correct answer is Option D .

Note:
Objects that undergo projectile motion always have a constant velocity in the horizontal direction and the vertical velocity changes constantly. The resultant trajectory always forms the shape of a parabola. It emphasises the fact that constant acceleration that is essentially unidirectional, is capable of producing two dimensional motion as the initial velocity and the force are in different directions.