A ship gun fires a 17 ton shell. Its canons max elevation is 50⁰ the shell having no air resistance was fired at an angle of 25⁰ the shells range is 60miles if fired at 50⁰ what is the shells max range if fired at 30⁰ . The speed of the shell is 2 miles per second the initial velocity of the shell after being fired is 2.6 miles per second

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.

Learn how to calculate the maximum range of a projectile fired at different angles. Solve for the range at 30 degrees using the given initial velocity and gravitational constant. Detailed step-by-step solution included.

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