A gun is mounted on a gun carriage movable on a smooth plane the gun is elevated at an angle `alpha` to the horizontal . A shot, fired leaves the gun in a direction inclined at an angle `theta` to the horizontal . If the mass of the gum and its carriage be n times that of short , find `tan theta : tan alpha` .

Explanation

To solve the problem, we need to analyze the motion of the gun and the projectile fired from it. We will use the principles of momentum conservation and the relationships between the angles and velocities involved.

### Step-by-step Solution:

1. **Identify the Masses**:
- Let the mass of the shot be \( m \).
- The mass of the gun and its carriage is \( n \times m \).

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