Animated Solution for Physics - Kinematics: Two guns A and B can fire bullets at speeds $1\text{ km/s}$ and $2\text{ km/s}$, respectively. From a point on a horizontal ground, they are fired in all possible directions. The ratio of maximum areas covered by the bullets on the ground fired by the two guns is

Imagine standing in the center of a vast, flat plain. You hold a gun and begin firing bullets in every possible horizontal direction. Where is it safe to stand? The bullets will land all around you, creating a circular "danger zone." The boundary of this zone is determined by the absolute maximum distance a bullet can travel before hitting the ground.

This maximum distance is what we call the maximum range of a projectile. Because you are firing in all $360^{\circ }$ around you, the locus of all these maximum reach points forms a perfect circle.

To find the radius of this circular zone, we need to recall the kinematics of projectile motion. The range $R$ of a projectile fired with an initial speed $u$ at an angle $\theta$ is given by:

To maximize this range, we must fire at an angle where $\sin (2\theta )=1$, which occurs at $\theta =45^{\circ }$. Substituting this in, we get the formula for the maximum range:

This $R_{max}$ is exactly the radius of our circular danger zone.

Now, the question asks for the ratio of the maximum areas covered by the two guns. The area $A$ of a circle is $\pi R^2$. Let's substitute our maximum range into the area formula:

Look closely at this result! The area covered is not proportional to the speed, nor is it proportional to the square of the speed. The area is proportional to the fourth power of the initial speed ($A\propto u^4$). This is a profound non-linear relationship. A small increase in speed will result in a massive expansion of the area covered.

We are given two guns: Gun A with speed $u_A=1\text{ km/s}$ and Gun B with speed $u_B=2\text{ km/s}$. We need to find the ratio of their areas, $\frac{A_A}{A_B}$. Since $\pi$ and $g$ are constants, they cancel out perfectly:

Substituting the given speeds:

Even though Gun B is only twice as fast as Gun A, it covers sixteen times the area! This beautifully illustrates how compounding exponents in physics can lead to dramatic real-world effects. The correct ratio is $1:16$.