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

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.

Calculate the maximum range of a projectile fired at a 30-degree angle with an initial speed of 2 miles per second. Learn the projectile motion formula and understand how angle affects range. Suitable for advanced high school students and beyond.

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