Let's solve the problem step by step. We are given the following information:

• The shell is fired at an initial speed of 2.6 miles per second.
• The shell was fired at an angle of 25° and had a range of 60 miles.
• We need to find the range when the shell is fired at 30°.

Step 1: Establish the relationship

The range $R$ of a projectile fired at an angle $\theta$ with an initial velocity $v$ is given by: $R = \frac{v^2 \sin(2\theta)}{g}$ where $g$ is the acceleration due to gravity. We can use this equation to determine the range for different angles.

Step 2: Find the effective gravitational constant

Since we know the range at 25°, we can use that to determine the effective value of $g$ in the given unit system.

Given: $R_{25} = 60 \text{ miles}$ $v = 2.6 \text{ miles/second}$ $\theta = 25^\circ$

Using the range formula: $60 = \frac{(2.6)^2 \sin(2 \times 25^\circ)}{g}$ $\sin(50^\circ) \approx 0.766$ $60 = \frac{6.76 \times 0.766}{g}$ $60 = \frac{5.17816}{g}$ $g = \frac{5.17816}{60}$ $g \approx 0.0863 \text{ miles/second}^2$

Step 3: Calculate the range at 30°

Now, we use this value of $g$ to calculate the range when the shell is fired at 30°.

Using: $v = 2.6 \text{ miles/second}$ $\theta = 30^\circ$ $R_{30} = \frac{(2.6)^2 \sin(2 \times 30^\circ)}{0.0863}$ $\sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866$ $R_{30} = \frac{6.76 \times 0.866}{0.0863}$ $R_{30} = \frac{5.85416}{0.0863}$ $R_{30} \approx 67.86 \text{ miles}$

Therefore, the maximum range if the shell is fired at an angle of 30° with an initial velocity of 2.6 miles per second is approximately 67.86 miles.

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

Here are five questions you might want to ask next:

1. How would changing the initial velocity affect the range of the projectile?
2. What would be the range if the shell were fired at the maximum elevation angle of 50°?
3. Can you explain how the range changes with different angles using a graph?
4. What is the time of flight for the shell fired at 30°?
5. How can we incorporate air resistance into these calculations?

Tip: Always ensure that units are consistent when using physics equations to avoid calculation errors.