Image of an object approaching a convex mirror of radius of curvature 20m slong its optical axis is observed to move from `(25)/(3)`m to `(50)/(7)`m in 30 seconds. What is the speed of the object in km per hour?

Explanation

To solve the problem step by step, we will use the mirror formula and the information provided about the image positions.

### Step 1: Understand the given data
- Radius of curvature (R) = 20 m
- Focal length (F) = R/2 = 20/2 = 10 m (for a convex mirror, F is positive)
- Initial image position (V1) = 25/3 m
- Final image position (V2) = 50/7 m
- Time taken (t) = 30 seconds

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

#### For the first position (V1):
1. Substitute V1 into the mirror formula:
\[ \frac{1}{10} = \frac{3}{25} + \frac{1}{u_1} \]
2. Rearranging gives:
\[ \frac{1}{u_1} = \frac{1}{10} - \frac{3}{25} \]
3. Finding a common denominator (50):
\[ \frac{1}{u_1} = \frac{5}{50} - \frac{6}{50} = -\frac{1}{50} \]
4. Therefore,
\[ u_1 = -50 \text{ m} \]

#### For the second position (V2):
1. Substitute V2 into the mirror formula:
\[ \frac{1}{10} = \frac{7}{50} + \frac{1}{u_2} \]
2. Rearranging gives:
\[ \frac{1}{u_2} = \frac{1}{10} - \frac{7}{50} \]
3. Finding a common denominator (50):
\[ \frac{1}{u_2} = \frac{5}{50} - \frac{7}{50} = -\frac{2}{50} \]
4. Therefore,
\[ u_2 = -25 \text{ m} \]

### Step 3: Calculate the distance covered by the object
- Initial position of the object: \( u_1 = -50 \) m
- Final position of the object: \( u_2 = -25 \) m
- Distance covered = \( |u_2 - u_1| = |-25 - (-50)| = |-25 + 50| = 25 \) m

### Step 4: Calculate the speed of the object
- Speed \( v \) is given by:
\[ v = \frac{\text{Distance covered}}{\text{Time}} = \frac{25 \text{ m}}{30 \text{ s}} = \frac{5}{6} \text{ m/s} \]

### Step 5: Convert speed to km/h
- To convert m/s to km/h, multiply by 3.6:
\[ v = \frac{5}{6} \times 3.6 = 3 \text{ km/h} \]

### Final Answer
The speed of the object is **3 km/h**.

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