A small block of mass m and a concave mirror of radius R fitted with a stand lie on a smooth horizontal table with a separation d between them. The mirror together with its stand has a mass m. The block is pushed at t = 0 towards the mirror so that it starts moving towards the mirror at a constant speed V and collides with it. The collision is perfectly elastic. Find the velocity of the image (a) at a time `t lt d/V,` (b) at a time `t gt d/V`.

Explanation

To solve the problem step by step, we need to analyze the motion of the block and the mirror, as well as the properties of the concave mirror.

### Step 1: Understanding the Initial Setup
We have a block of mass \( m \) and a concave mirror with radius \( R \) on a smooth horizontal table, separated by a distance \( d \). The block is moving towards the mirror with a constant speed \( V \).

### Step 2: Determine Object Distance Before Collision
At time \( t \), the distance of the block from the mirror can be expressed as:
\[ u = - (d - Vt) \]
This is because the object distance \( u \) is taken as negative in the mirror formula convention, and the block is moving towards the mirror.

### Step 3: Calculate Focal Length of the Mirror
The focal length \( f \) of a concave mirror is given by:
\[ f = -\frac{R}{2} \]

### Step 4: Apply the Mirror Formula
Using the mirror formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values of \( f \) and \( u \):
\[ \frac{1}{-\frac{R}{2}} = \frac{1}{v} + \frac{1}{-(d - Vt)} \]
This simplifies to:
\[ -\frac{2}{R} = \frac{1}{v} - \frac{1}{(d - Vt)} \]

### Step 5: Solve for Image Distance \( v \)
Rearranging gives:
\[ \frac{1}{v} = -\frac{2}{R} + \frac{1}{(d - Vt)} \]
\[ \frac{1}{v} = \frac{(d - Vt) - (-2R)}{R(d - Vt)} \]
\[ v = \frac{R(d - Vt)}{(d - Vt) + 2R} \]

### Step 6: Find the Velocity of the Image for \( t < \frac{d}{V} \)
For \( t < \frac{d}{V} \), the block has not yet collided with the mirror. The velocity of the image \( v_i \) can be found by differentiating \( v \) with respect to time \( t \):
\[ \frac{dv}{dt} = \frac{d}{dt} \left( \frac{R(d - Vt)}{(d - Vt) + 2R} \right) \]
Using the quotient rule, we can find \( \frac{dv}{dt} \).

### Step 7: Collision and Post-Collision Analysis
When the block collides with the mirror, since the collision is perfectly elastic, the block will come to rest, and the mirror will move with velocity \( V \).

### Step 8: Determine Object Distance After Collision
After the collision, the object distance \( u \) becomes:
\[ u = d - Vt \]
for \( t > \frac{d}{V} \).

### Step 9: Apply the Mirror Formula Again
Using the same mirror formula:
\[ \frac{1}{v} = -\frac{2}{R} + \frac{1}{(d - Vt)} \]
We can solve for \( v \) again.

### Step 10: Final Velocity of the Image
The final velocity of the image after the collision can be expressed as:
\[ v_i = \text{velocity of the mirror} + \text{velocity of the image due to the mirror} \]

### Summary of Results
- For \( t < \frac{d}{V} \): The velocity of the image can be calculated using the derived formula.
- For \( t > \frac{d}{V} \): The velocity of the image will be influenced by the motion of the mirror and can be calculated similarly.