A large number of bullets are fired in all directions with the same speed `v`. Find the maximum area on the ground on which these bullets will spread.

Options

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

Explanation

To solve the problem of finding the maximum area on the ground where bullets fired in all directions will spread, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Bullets are fired from a point in all directions with the same speed `v`.
- The bullets will spread out and form a circular area on the ground.

2. **Identifying the Radius of the Circular Area**:
- The radius of the circular area (which is also the range of the bullets) can be determined using the formula for the range of a projectile:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
- Here, `u` is the initial speed (which is `v`), `g` is the acceleration due to gravity, and `θ` is the angle of projection.

3. **Maximizing the Range**:
- To find the maximum range, we need to maximize the term `sin 2θ`.
- The maximum value of `sin 2θ` is 1, which occurs when `2θ = 90°` or `θ = 45°`.

4. **Calculating Maximum Range**:
- Substituting `sin 2θ = 1` into the range formula gives:
\[ R_{\text{max}} = \frac{v^2 \cdot 1}{g} = \frac{v^2}{g} \]

5. **Finding the Area of the Circle**:
- The area `A` of a circle is given by the formula:
\[ A = \pi r^2 \]
- Here, `r` is the radius, which we have found to be the maximum range:
\[ A = \pi \left(\frac{v^2}{g}\right)^2 \]

6. **Simplifying the Area Expression**:
- Simplifying the expression for the area gives:
\[ A = \pi \frac{v^4}{g^2} \]

7. **Final Result**:
- Therefore, the maximum area on the ground on which the bullets will spread is:
\[ A = \frac{\pi v^4}{g^2} \]

### Summary:
The maximum area on the ground where the bullets will spread is given by:
\[ A = \frac{\pi v^4}{g^2} \]