To solve this problem, we need to find the range of a projectile fired at a given angle with a specific initial velocity. We'll use the projectile motion equations, taking into account the initial speed and angle of launch. Here are the steps involved:

Step-by-Step Solution

1. Initial Velocity (v): $v = 2 \text{ miles/second}$

2. Initial Angle (θ):

   • Given max elevation: $\theta_{\text{max}} = 50^\circ$
   • Given firing angle: $\theta = 30^\circ$

3. Range Formula: The range $R$ of a projectile fired at an angle $\theta$ with initial speed $v$ is given by: $R = \frac{v^2 \sin(2\theta)}{g}$ where $g$ is the acceleration due to gravity. Assuming $g \approx 9.8 \, \text{m/s}^2$, we need to convert it to miles per second squared for consistency.

   Since $1 \, \text{mile} = 1609.34 \, \text{meters}$: $g \approx 9.8 \, \text{m/s}^2 \times \left( \frac{1 \, \text{mile}}{1609.34 \, \text{meters}} \right)^2 \approx 0.00000303 \, \text{miles/second}^2$

4. Calculating the Range:

   • For $\theta = 30^\circ$: $R_{30} = \frac{(2 \, \text{miles/second})^2 \sin(2 \times 30^\circ)}{0.00000303 \, \text{miles/second}^2}$ $\sin(60^\circ) = \frac{\sqrt{3}}{2}$ $R_{30} = \frac{4 \times \frac{\sqrt{3}}{2}}{0.00000303}$ $R_{30} = \frac{2\sqrt{3}}{0.00000303}$ $R_{30} = \frac{2 \times 1.732}{0.00000303}$ $R_{30} = \frac{3.464}{0.00000303} \approx 1,143,564.7 \text{ miles}$

Hence, the maximum range if fired at $30^\circ$ is approximately $1,143,564.7 \text{ miles}$.

Summary

The shell's maximum range if fired at an angle of $30^\circ$ with an initial speed of $2 \text{ miles/second}$ is approximately 1,143,564.7 miles.

Would you like details on any specific step or have any questions?

Here are five questions you might want to ask next:

1. How does the range of a projectile change with different angles of launch?
2. Can you explain the significance of the angle that maximizes the range of a projectile?
3. How would air resistance affect the range of the projectile?
4. Can we calculate the time of flight for the shell fired at 30°?
5. How do different initial velocities affect the projectile's range?

Tip: For maximum range in projectile motion without air resistance, the optimal launch angle is 45 degrees.