Several bullets are fired from a particular gun in various possible directions . If bullets are fired with a speed u, then maximum area of ground covered by the bullets is

Options

A. `(pi u^(4))/(g)`
B. `(pi u^(4))/(g^(2))`
C. `(4 pi u^(4))/(g^(2))`
D. `(pi u ^(4))/(g^(4))`

Explanation

To solve the problem of finding the maximum area of ground covered by bullets fired from a gun at speed \( u \), we can follow these steps:

### Step 1: Understand the Motion of the Bullets
When bullets are fired from a gun, they follow a projectile motion. The range of a projectile is the horizontal distance it covers before hitting the ground. The bullets can be fired in various directions, meaning we need to find the maximum range achievable by the bullets.

**Hint:** Remember that the range of a projectile depends on the angle of projection and the initial speed.

### Step 2: Determine the Maximum Range
The range \( R \) of a projectile launched with an initial speed \( u \) at an angle \( \theta \) is given by the formula:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
To maximize the range, we need to maximize \( \sin(2\theta) \). The maximum value of \( \sin(2\theta) \) is 1, which occurs when \( 2\theta = 90^\circ \) or \( \theta = 45^\circ \).

**Hint:** What angle gives the maximum value for the sine function?

### Step 3: Calculate the Maximum Range
Substituting \( \sin(2\theta) = 1 \) into the range formula, we get:
\[ R_{\text{max}} = \frac{u^2}{g} \]

**Hint:** Make sure to keep track of the units when substituting values.

### Step 4: Find the Area Covered
The area \( A \) covered by the bullets can be visualized as a circle with the maximum range \( R_{\text{max}} \) as its radius. The area of a circle is given by:
\[ A = \pi R^2 \]
Substituting \( R_{\text{max}} \) into the area formula:
\[ A = \pi \left(\frac{u^2}{g}\right)^2 = \pi \frac{u^4}{g^2} \]

**Hint:** Remember the formula for the area of a circle and how to substitute the radius.

### Step 5: Final Result
Thus, the maximum area of the ground covered by the bullets is:
\[ A = \frac{\pi u^4}{g^2} \]

**Hint:** Ensure that you understand the relationship between the speed of the bullets and the area covered.

### Conclusion
The maximum area of ground covered by the bullets fired from the gun is \( \frac{\pi u^4}{g^2} \).