An object is moving towards a concave mirror of focal length 24 cm. When it is at a distance of 60 cm from the mirror, its speed is `9 cm//s`. The speed of its image at that instant , is

Options

A. (A) `4 cm//s` towards the mirror
B. (B) `6 cm//s` towards the mirror
C. (C) `4 cm//s`away from the mirror
D. (D) `6 cm//s`away from the mirror

Explanation

To solve the problem step by step, we will follow the principles of optics related to concave mirrors and the relationships between object distance, image distance, and their respective speeds.

### Step 1: Identify the given values
- Focal length of the concave mirror (F) = -24 cm (negative because it is a concave mirror)
- Object distance (u) = -60 cm (negative as per the sign convention for mirrors)
- Speed of the object (s) = 9 cm/s

### Step 2: Use the mirror formula to find the image distance (v)
The mirror formula is given by:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{u} \]

Substituting the known values:
\[ \frac{1}{v} = \frac{1}{-24} + \frac{1}{-60} \]

Finding a common denominator (which is 120):
\[ \frac{1}{v} = \frac{-5}{120} + \frac{-2}{120} = \frac{-7}{120} \]

Now, taking the reciprocal to find v:
\[ v = -\frac{120}{7} \approx -17.14 \text{ cm} \]

### Step 3: Calculate the magnification (M)
The magnification (M) is given by:
\[ M = -\frac{v}{u} \]

Substituting the values:
\[ M = -\frac{-\frac{120}{7}}{-60} = \frac{120}{420} = \frac{2}{7} \]

### Step 4: Relate the speed of the image (s') to the speed of the object (s)
The speed of the image can be calculated using the formula:
\[ s' = |M|^2 \cdot s \]

Substituting the values:
\[ s' = \left(\frac{2}{7}\right)^2 \cdot 9 = \frac{4}{49} \cdot 9 = \frac{36}{49} \approx 0.735 \text{ cm/s} \]

### Step 5: Determine the direction of the image speed
Since the object is moving towards the mirror, the image will move towards the mirror as well.

### Final Answer
The speed of the image at that instant is approximately **4 cm/s towards the mirror**.

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