A police inspector is chasing a thief who is running away in a car with a speed 3m/s. The speed of police jeep is 12 m/s. Then the speed of image of police jeep as seen by thief in the rear view mirror when the police jeep is at a distance of 30 m is (value of focal length of the rear view mirror is 15 m)

Options

A. 2m/s
B. 3m/s
C. 4m/s
D. 1m/s

Answer

To solve the problem step by step, we will follow the sequence of calculations as outlined in the video transcript.

### Step 1: Identify the parameters
- Speed of the thief (car): \( v_t = 3 \, \text{m/s} \)
- Speed of the police jeep: \( v_p = 12 \, \text{m/s} \)
- Distance of the police jeep from the thief: \( u = -30 \, \text{m} \) (negative because it is on the same side as the object)
- Focal length of the rear view mirror (convex mirror): \( f = -15 \, \text{m} \)

### Step 2: Use the mirror formula
The mirror formula for a convex mirror is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values:
\[ \frac{1}{-15} = \frac{1}{v} + \frac{1}{-30} \]

### Step 3: Solve for image distance \( v \)
Rearranging the equation:
\[ \frac{1}{v} = \frac{1}{-15} + \frac{1}{30} \]
Finding a common denominator (which is 30):
\[ \frac{1}{v} = \frac{-2}{30} + \frac{1}{30} = \frac{-1}{30} \]
Thus,
\[ v = -30 \, \text{m} \]

### Step 4: Differentiate to find the speed of the image
We need to find the speed of the image as seen by the thief. We will differentiate the mirror formula with respect to time \( t \):
\[ -\frac{1}{f^2} \frac{df}{dt} = -\frac{1}{v^2} \frac{dv}{dt} - \frac{1}{u^2} \frac{du}{dt} \]
Since the focal length \( f \) is constant, \( \frac{df}{dt} = 0 \):
\[ 0 = -\frac{1}{v^2} \frac{dv}{dt} - \frac{1}{u^2} \frac{du}{dt} \]

### Step 5: Calculate the relative speed
The relative speed of the police jeep with respect to the thief is:
\[ v_{om} = v_p - v_t = 12 - 3 = 9 \, \text{m/s} \]
Substituting \( u = -30 \, \text{m} \) and \( v = -10 \, \text{m} \):
\[ \frac{dv}{dt} = \frac{v^2}{u^2} v_{om} \]
Substituting the values:
\[ \frac{dv}{dt} = \frac{(-10)^2}{(-30)^2} \cdot 9 = \frac{100}{900} \cdot 9 = \frac{1}{9} \cdot 9 = 1 \, \text{m/s} \]

### Step 6: Calculate the speed of the image
The speed of the image \( v_i \) as seen by the thief is:
\[ v_i = v_{om} + v_t = 1 + 3 = 4 \, \text{m/s} \]

### Final Answer
The speed of the image of the police jeep as seen by the thief in the rear view mirror is **4 m/s**.

---