A real 1 mm long object is kept perpendicular to the principal axis of a concave mirror. The size of the image formed is 1/4 mm. Now the mirror starts moving, away, from the object with a velocity 2 cm/s along the principal axis.

Options

A. the velocity of the image along the principal axis at the given instant will be `(17)/(8) cm//s` towards the mirror
B. the length of the image will decrease as the mirror moves
C. the speed of the image will always be more than `2 cm//s`
D. the rate with which the length of the image will change is a constant

Explanation

To solve the problem step by step, we will analyze the situation involving the concave mirror, the object, and the image formed.

### Step 1: Understanding the Given Information
- The length of the object (h_o) = 1 mm
- The length of the image (h_i) = 1/4 mm
- The mirror is moving away from the object with a velocity (v_m) = 2 cm/s.

### Step 2: Calculate the Magnification
The magnification (m) of the mirror is given by the formula:
\[ m = \frac{h_i}{h_o} \]

Substituting the values:
\[ m = \frac{1/4 \text{ mm}}{1 \text{ mm}} = \frac{1}{4} \]

### Step 3: Determine the Velocity of the Image
The velocity of the image (v_i) can be determined using the magnification and the velocity of the mirror. The formula relating the velocities is:
\[ v_i = m^2 \cdot v_m \]

Substituting the values:
\[ v_i = \left(\frac{1}{4}\right)^2 \cdot 2 \text{ cm/s} \]
\[ v_i = \frac{1}{16} \cdot 2 \text{ cm/s} = \frac{2}{16} \text{ cm/s} = \frac{1}{8} \text{ cm/s} \]

### Step 4: Determine the Direction of the Image Velocity
Since the mirror is moving away from the object, the image will move in the opposite direction to the movement of the mirror. Therefore, the image velocity is towards the right.

### Step 5: Calculate the Velocity of the Image with Respect to the Ground
The velocity of the image with respect to the ground (v_i') can be calculated as:
\[ v_i' = v_i + v_m \]
Where:
- \( v_i \) is the velocity of the image towards the right (positive direction).
- \( v_m \) is the velocity of the mirror moving away from the object (negative direction).

Substituting the values:
\[ v_i' = \frac{1}{8} \text{ cm/s} + 2 \text{ cm/s} \]
\[ v_i' = \frac{1}{8} + \frac{16}{8} = \frac{17}{8} \text{ cm/s} \]

### Step 6: Conclusion
The velocity of the image with respect to the ground is \( \frac{17}{8} \text{ cm/s} \) towards the mirror.