A gun of mass `m_1` fires a bullet of mass `m_2` with a horizontal speed
` v_0`. The gun is fitted with a concave mirror of focal length f facing towards a
receding bullet.Find the speed of separations of the bullet and the image just after the gun was fired.

Explanation

To solve the problem step by step, we will analyze the situation involving the gun, bullet, and concave mirror.

### Step 1: Understand the system
We have a gun of mass \( m_1 \) that fires a bullet of mass \( m_2 \) with a horizontal speed \( v_0 \). The gun is fitted with a concave mirror of focal length \( f \) facing the bullet.

### Step 2: Apply the conservation of momentum
When the bullet is fired, the total momentum of the system (gun + bullet) must be conserved. Initially, both the gun and bullet are at rest, so the total initial momentum is zero.

After firing, let the velocity of the gun be \( v_1 \) (in the opposite direction of the bullet). The momentum conservation equation can be written as:
\[ m_2 v_0 - m_1 v_1 = 0 \]
From this, we can express the velocity of the gun:
\[ m_1 v_1 = m_2 v_0 \implies v_1 = \frac{m_2 v_0}{m_1} \]

### Step 3: Determine the speed of the image
For a concave mirror, the mirror formula relates the object distance \( u \), image distance \( v \), and focal length \( f \) as follows:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Differentiating this with respect to time \( t \) gives us the relationship between the velocities of the object and the image. The velocity of the image \( v_i \) can be expressed as:
\[ v_i = -\frac{v^2}{u^2} v_o \]
where \( v_o \) is the velocity of the object (the bullet).

### Step 4: Substitute the values
Since the bullet is moving with speed \( v_0 \) and just after firing, the distance \( u \) can be approximated as the distance to the mirror. For our calculations, we can assume \( u \) is effectively \( v_0 \) since the bullet is moving away from the mirror. Thus:
\[ v_i = -\frac{v_0^2}{v_0^2} v_0 = -v_0 \]
This indicates that the image moves in the opposite direction to the bullet.

### Step 5: Calculate the speed of separation
The speed of separation between the bullet and its image is the sum of their speeds since they are moving in opposite directions:
\[ \text{Speed of separation} = v_0 + |v_i| = v_0 + v_0 = 2v_0 \]

### Final Answer
Thus, the speed of separation of the bullet and the image just after the gun was fired is:
\[ \text{Speed of separation} = 2v_0 \]

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