An object moving at a speed of 5m/s towards a concave mirror of focal length f=1 m is at a distance of 9m. The average speed of the image is

Options

A. 1/5 m/s
B. 1/10 m/s
C. 5/9 m/s
D. 2/8 m/s

Explanation

To find the average speed of the image formed by a concave mirror when an object is moving towards it, we can follow these steps:

### Step 1: Understand the given data
- Speed of the object (u) = 5 m/s (towards the mirror)
- Initial distance of the object from the mirror (u_initial) = -9 m (negative because it is measured against the direction of the incident light)
- Focal length of the concave mirror (f) = -1 m (negative for concave mirrors)

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

### Step 3: Calculate the initial image distance (v_initial)
Substituting the values into the mirror formula:
\[ \frac{1}{v} + \frac{1}{-9} = \frac{1}{-1} \]
This simplifies to:
\[ \frac{1}{v} - \frac{1}{9} = -1 \]
\[ \frac{1}{v} = -1 + \frac{1}{9} \]
\[ \frac{1}{v} = -\frac{9}{9} + \frac{1}{9} = -\frac{8}{9} \]
Thus,
\[ v_{initial} = -\frac{9}{8} \text{ m} \]

### Step 4: Calculate the final object distance (u_final)
After 1 second, the object moves 5 m towards the mirror:
\[ u_{final} = -9 \text{ m} + 5 \text{ m} = -4 \text{ m} \]

### Step 5: Calculate the final image distance (v_final)
Using the mirror formula again for the new object distance:
\[ \frac{1}{v} + \frac{1}{-4} = \frac{1}{-1} \]
This simplifies to:
\[ \frac{1}{v} - \frac{1}{4} = -1 \]
\[ \frac{1}{v} = -1 + \frac{1}{4} \]
\[ \frac{1}{v} = -\frac{4}{4} + \frac{1}{4} = -\frac{3}{4} \]
Thus,
\[ v_{final} = -\frac{4}{3} \text{ m} \]

### Step 6: Calculate the change in image distance (Δv)
\[ \Delta v = v_{final} - v_{initial} = -\frac{4}{3} - \left(-\frac{9}{8}\right) \]
Finding a common denominator (24):
\[ \Delta v = -\frac{32}{24} + \frac{27}{24} = -\frac{5}{24} \text{ m} \]

### Step 7: Calculate the average speed of the image
The average speed of the image is given by the change in image distance over time:
\[ \text{Average Speed} = \frac{\Delta v}{\Delta t} = \frac{-\frac{5}{24}}{1 \text{ s}} = -\frac{5}{24} \text{ m/s} \]
Since speed is a scalar quantity, we take the absolute value:
\[ \text{Average Speed} = \frac{5}{24} \text{ m/s} \approx 0.208 \text{ m/s} \]

### Step 8: Final answer
The average speed of the image is approximately:
\[ \frac{1}{5} \text{ m/s} \text{ (as per the options given)} \]

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