A gun is fired from a moving platform and ranges of the shot are observed to be `R_1 and R_2` when the platform is moving forwards and backwards, respectively, with velocity `v_P`. Find the elevation of the gun `prop` in terms of the given quantities.

Explanation

To solve the problem, we need to analyze the motion of the projectile fired from a moving platform. Let's break it down step by step.

### Step 1: Define the Variables
Let:
- \( u \) = initial horizontal component of the velocity of the projectile with respect to the platform.
- \( v \) = initial vertical component of the velocity of the projectile with respect to the platform.
- \( v_P \) = velocity of the platform.
- \( R_1 \) = range of the shot when the platform is moving forwards.
- \( R_2 \) = range of the shot when the platform is moving backwards.
- \( g \) = acceleration due to gravity.

### Step 2: Write the Range Equations
When the gun is fired from the moving platform:
1. For the forward motion:
\[ R_1 = \frac{2(u + v_P)v}{g} \]
2. For the backward motion:
\[ R_2 = \frac{2(u - v_P)v}{g} \]

### Step 3: Add and Subtract the Range Equations
Now, we can add and subtract these two equations to eliminate \( u \):
- Adding:
\[ R_1 + R_2 = \frac{2(u + v_P)v}{g} + \frac{2(u - v_P)v}{g} \]
This simplifies to:
\[ R_1 + R_2 = \frac{4uv}{g} \]

- Subtracting:
\[ R_1 - R_2 = \frac{2(u + v_P)v}{g} - \frac{2(u - v_P)v}{g} \]
This simplifies to:
\[ R_1 - R_2 = \frac{4v_Pv}{g} \]

### Step 4: Square the Difference of Ranges
Now, we square the difference of the ranges:
\[ (R_1 - R_2)^2 = \left(\frac{4v_Pv}{g}\right)^2 = \frac{16v_P^2v^2}{g^2} \]

### Step 5: Express \( v \) in Terms of \( R_1 \) and \( R_2 \)
From the previous equations, we can express \( v \) in terms of \( R_1 \) and \( R_2 \):
\[ v^2 = \frac{g^2(R_1 - R_2)^2}{16v_P^2} \]

### Step 6: Substitute \( v \) into the Range Equation
Now, substitute \( v \) back into the equation for \( R_1 + R_2 \):
\[ R_1 + R_2 = \frac{4u}{g} \cdot \sqrt{\frac{g^2(R_1 - R_2)^2}{16v_P^2}} \]
This simplifies to:
\[ R_1 + R_2 = \frac{4u(R_1 - R_2)}{4v_P} \Rightarrow R_1 + R_2 = \frac{u(R_1 - R_2)}{v_P} \]

### Step 7: Solve for the Elevation Angle
The elevation angle \( \alpha \) can be expressed as:
\[ \tan \alpha = \frac{v}{u} \]
Using the expressions derived earlier, we can substitute:
\[ \tan \alpha = \frac{g}{4v_P^2} \cdot \frac{R_1 - R_2}{R_1 + R_2} \]

### Final Expression for Elevation
Thus, the elevation of the gun \( \alpha \) is given by:
\[ \alpha = \tan^{-1} \left( \frac{g}{4v_P^2} \cdot \frac{R_1 - R_2}{R_1 + R_2} \right) \]