An object is 20 cm away from a concave mirror with focal length 15 cm. If the object moves with a speed of 5 m/s along the axis, then the speed of the image will be

Options

A. 45 m/s
B. 27 m/s
C. 9 m/s
D. 10 m/s

Explanation

To solve the problem step by step, we will use the mirror formula and the concept of differentiation to find the speed of the image when the object moves.

### Step 1: Identify Given Values
- Object distance (u) = -20 cm (negative because it is in front of the mirror)
- Focal length (f) = -15 cm (negative for concave mirrors)
- Speed of the object (du/dt) = 5 m/s = 500 cm/s (conversion from m/s to cm/s)

### Step 2: Use the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{-15} = \frac{1}{v} + \frac{1}{-20} \]

### Step 3: Solve for Image Distance (v)
Rearranging the equation:
\[ \frac{1}{v} = \frac{1}{-15} + \frac{1}{20} \]
Finding a common denominator (60):
\[ \frac{1}{v} = \frac{-4}{60} + \frac{3}{60} = \frac{-1}{60} \]
Thus,
\[ v = -60 \text{ cm} \]
(The negative sign indicates that the image is formed on the same side as the object, which is typical for concave mirrors.)

### Step 4: Differentiate the Mirror Formula
Differentiating the mirror formula with respect to time (t):
\[ 0 = -\frac{1}{v^2} \frac{dv}{dt} - \frac{1}{u^2} \frac{du}{dt} \]
Rearranging gives:
\[ \frac{dv}{dt} = -\frac{v^2}{u^2} \frac{du}{dt} \]

### Step 5: Substitute Known Values
Substituting \(v = -60 \text{ cm}\), \(u = -20 \text{ cm}\), and \(\frac{du}{dt} = 500 \text{ cm/s}\):
\[ \frac{dv}{dt} = -\frac{(-60)^2}{(-20)^2} \times 500 \]
Calculating:
\[ \frac{dv}{dt} = -\frac{3600}{400} \times 500 = -9 \times 500 = -4500 \text{ cm/s} \]
Thus, the speed of the image is:
\[ \frac{dv}{dt} = 4500 \text{ cm/s} = 45 \text{ m/s} \]

### Final Answer
The speed of the image is **45 m/s**.

---