A particle is moving at a constant speed V from a large distance towards a concave mirror of radius R along its principal axis. Find the speed of the image formed by the mirror as a function of the distance x of the particle from the mirror.

Explanation

To solve the problem of finding the speed of the image formed by a concave mirror as a function of the distance \( x \) of a particle from the mirror, let's break it down step by step.

### Step 1: Identify Given Information
- The radius of the concave mirror is \( R \).
- The focal length \( f \) of the concave mirror is given by:
\[ f = \frac{R}{2} \]
- The particle is moving towards the mirror at a constant speed \( V \).

### Step 2: Define the Object Distance
- The object distance \( u \) is negative for a concave mirror, so:
\[ u = -x \]

### Step 3: Use the Mirror Formula
The mirror formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f \):
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
Substituting the values we have:
\[ \frac{1}{-\frac{R}{2}} = \frac{1}{-x} + \frac{1}{v} \]
This simplifies to:
\[ -\frac{2}{R} = -\frac{1}{x} + \frac{1}{v} \]

### Step 4: Rearranging the Equation
Rearranging the equation gives:
\[ \frac{1}{v} = -\frac{2}{R} + \frac{1}{x} \]
This can be rewritten as:
\[ \frac{1}{v} = \frac{x - \frac{2R}{x}}{x} \]
Thus:
\[ v = \frac{Rx}{R - 2x} \]

### Step 5: Differentiate to Find the Speed of the Image
Let \( v_1 \) be the speed of the image. We need to differentiate \( v \) with respect to time \( t \):
\[ v_1 = \frac{d}{dt}\left(\frac{Rx}{R - 2x}\right) \]
Using the quotient rule:
\[ v_1 = \frac{(R - 2x) \cdot \frac{d}{dt}(Rx) - Rx \cdot \frac{d}{dt}(R - 2x)}{(R - 2x)^2} \]

### Step 6: Calculate Derivatives
Calculating the derivatives:
- \( \frac{d}{dt}(Rx) = R \frac{dx}{dt} = R V \) (since \( \frac{dx}{dt} = V \))
- \( \frac{d}{dt}(R - 2x) = -2 \frac{dx}{dt} = -2V \)

Substituting these into our expression for \( v_1 \):
\[ v_1 = \frac{(R - 2x)(RV) - Rx(-2V)}{(R - 2x)^2} \]

### Step 7: Simplify the Expression
This simplifies to:
\[ v_1 = \frac{(R - 2x)RV + 2R x V}{(R - 2x)^2} \]
Factoring out \( V \):
\[ v_1 = V \cdot \frac{R^2}{(R - 2x)^2} \]

### Final Result
Thus, the speed of the image formed by the mirror as a function of the distance \( x \) of the particle from the mirror is:
\[ v_1 = V \cdot \frac{R^2}{(R - 2x)^2} \]