Question

A large number of bullets are fired in all directions with the same speed \( v \). What is the maximum area on the ground on which these bullets will spread?

Hint:

The maximum spread area for projectiles depends on the square of their speed and is inversely proportional to the square of gravitational acceleration.

Answer 1

Step 1: Projectile Horizontal Range
The horizontal range \( R \) of a projectile is defined as:\[R = \frac{u^2 \sin 2\theta}{g}\]Here, \( u \) represents the initial speed, \( \theta \) is the angle of projection, and \( g \) denotes the acceleration due to gravity.Step 2: Maximum Horizontal Range
The maximum horizontal range is achieved when \( \sin 2\theta = 1 \), which occurs at \( \theta = 45^\circ \). This yields:\[R_{\text{max}} = \frac{u^2}{g}\]Step 3: Area of Spread
If projectiles are launched in all directions, their spread forms a circle with a radius equal to \( R_{\text{max}} \). The area \( A \) of this spread is calculated as:\[A = \pi R_{\text{max}}^2 = \pi \left( \frac{u^2}{g} \right)^2\]