A gum (mass=M) fires a bullet (mass=m) with speed `v_r` relative to barrel of the gum which is inclined at an anlge of `60^@` with horizontal. The gun is placed over a smooth horizontal surface. Find the recoil speed of gun.

Explanation

To solve the problem of finding the recoil speed of the gun when it fires a bullet, we can use the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Identify the system
We have a gun of mass \( M \) and a bullet of mass \( m \). The bullet is fired with a speed \( v_r \) relative to the barrel of the gun, which is inclined at an angle of \( 60^\circ \) with the horizontal.

### Step 2: Set up the momentum conservation equation
Since the gun is placed on a smooth horizontal surface, there are no external horizontal forces acting on the system. Therefore, the total momentum before firing must equal the total momentum after firing.

Before firing:
- The total momentum is \( 0 \) because both the gun and the bullet are at rest.

After firing:
- The momentum of the bullet can be resolved into horizontal and vertical components. The horizontal component of the bullet's velocity is given by:
\[ v_{b_x} = v_r \cos(60^\circ) = v_r \cdot \frac{1}{2} = \frac{v_r}{2} \]
- The momentum of the bullet is then \( m \cdot v_{b_x} = m \cdot \frac{v_r}{2} \).

Let \( v \) be the recoil speed of the gun. The momentum of the gun is \( -M \cdot v \) (the negative sign indicates that the gun moves in the opposite direction to the bullet).

### Step 3: Write the conservation of momentum equation
According to the conservation of momentum:
\[ \text{Total momentum before firing} = \text{Total momentum after firing} \]
This gives us:
\[ 0 = -M \cdot v + m \cdot \frac{v_r}{2} \]

### Step 4: Solve for the recoil speed \( v \)
Rearranging the equation gives:
\[ M \cdot v = m \cdot \frac{v_r}{2} \]
Thus, we can solve for \( v \):
\[ v = \frac{m \cdot \frac{v_r}{2}}{M} = \frac{m \cdot v_r}{2M} \]

### Final Answer
The recoil speed of the gun is:
\[ v = \frac{m \cdot v_r}{2M} \]