A thief is running away in a car with velocity of `20 m//s`. A police jeep is following him, which is sighted by thief in his rear view mirror, which is a convex mirror of focal length 10 m. He observes that the image of jeep is moving towards him with a velocity of `1 cm//s`. if the magnification of mirror for the jeep at that time is `1/10`. Find
(a) the actual speed of jeep,
(b) rate at which magnification is changing.
Assume the police's jeep is on the axis of the mirror.

Explanation

To solve the problem step by step, we will break it down into two parts: (a) finding the actual speed of the jeep and (b) determining the rate at which magnification is changing.

### Step-by-Step Solution:

#### Part (a): Finding the Actual Speed of the Jeep

1. **Understanding the Given Information**:
- Thief's speed (V_thief) = 20 m/s
- Image of the jeep is moving towards the thief with a velocity (V_image) = 1 cm/s = 0.01 m/s
- Magnification (m) = 1/10
- Focal length (f) = 10 m

2. **Using the Magnification Formula**:
The magnification (m) for a mirror is given by:
\[ m = \frac{V}{U} \]
where \( V \) is the image distance and \( U \) is the object distance. Given that \( m = \frac{1}{10} \), we can express:
\[ V = \frac{U}{10} \]

3. **Using the Mirror Formula**:
The mirror formula is:
\[ \frac{1}{f} = \frac{1}{U} + \frac{1}{V} \]
Substituting \( V = \frac{U}{10} \) into the mirror formula:
\[ \frac{1}{10} = \frac{1}{U} + \frac{10}{U} \]
This simplifies to:
\[ \frac{1}{10} = \frac{11}{U} \]
Rearranging gives:
\[ U = 110 \text{ m} \]

4. **Finding the Speed of the Jeep**:
The relative speed of the jeep with respect to the image can be expressed as:
\[ V_{relative} = m^2 \cdot V_{jeep} \]
where \( V_{jeep} \) is the speed of the jeep. Given \( V_{relative} = 0.01 \text{ m/s} \) and \( m = \frac{1}{10} \):
\[ 0.01 = \left(\frac{1}{10}\right)^2 \cdot V_{jeep} \]
\[ 0.01 = \frac{1}{100} \cdot V_{jeep} \]
Therefore:
\[ V_{jeep} = 0.01 \times 100 = 1 \text{ m/s} \]

5. **Calculating the Actual Speed of the Jeep**:
Since both the thief and the jeep are moving in the same direction, the actual speed of the jeep (V_actual) is:
\[ V_{actual} = V_{thief} + V_{jeep} = 20 \text{ m/s} + 1 \text{ m/s} = 21 \text{ m/s} \]

#### Part (b): Rate at Which Magnification is Changing

1. **Using the Rate of Change of Magnification Formula**:
The rate of change of magnification is given by:
\[ \frac{dm}{dt} = -\frac{f}{U^2} \frac{dU}{dt} \]
where \( \frac{dU}{dt} \) is the speed of the jeep (V_jeep = 1 m/s).

2. **Substituting the Known Values**:
We have:
- \( f = 10 \text{ m} \)
- \( U = 110 \text{ m} \)
- \( \frac{dU}{dt} = 1 \text{ m/s} \)

Substituting these values into the formula:
\[ \frac{dm}{dt} = -\frac{10}{(110)^2} \cdot 1 \]
\[ \frac{dm}{dt} = -\frac{10}{12100} = -\frac{1}{1210} \text{ per second} \approx -0.000826 \text{ per second} \]

### Final Answers:
(a) The actual speed of the jeep is **21 m/s**.
(b) The rate at which magnification is changing is approximately **-0.000826 per second**.