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 follow the same logical approach as outlined in the video transcript.

### Step 1: Understand the given data
- The radius of curvature (R) of the convex mirror is given as 20 m.
- The initial image position (V1) is \( \frac{25}{3} \) m.
- The final image position (V2) is \( \frac{50}{7} \) m.
- The time taken for this change is 30 seconds.

### Step 2: Calculate the focal length (f)
The focal length (f) of a mirror is given by:
\[ f = \frac{R}{2} \]
Substituting the value of R:
\[ f = \frac{20}{2} = 10 \text{ m} \]

### Step 3: Use the mirror formula to find object distances (u)
The mirror formula is:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
We will use this formula to find the object distance (u) for both image positions.

#### For the first image position (V1):
\[ V_1 = \frac{25}{3} \text{ m} \]
Substituting into the mirror formula:
\[ \frac{1}{10} = \frac{3}{25} + \frac{1}{u_1} \]
Rearranging gives:
\[ \frac{1}{u_1} = \frac{1}{10} - \frac{3}{25} \]
Finding a common denominator (50):
\[ \frac{1}{u_1} = \frac{5}{50} - \frac{6}{50} = -\frac{1}{50} \]
Thus:
\[ u_1 = -50 \text{ m} \]

#### For the second image position (V2):
\[ V_2 = \frac{50}{7} \text{ m} \]
Substituting into the mirror formula:
\[ \frac{1}{10} = \frac{7}{50} + \frac{1}{u_2} \]
Rearranging gives:
\[ \frac{1}{u_2} = \frac{1}{10} - \frac{7}{50} \]
Finding a common denominator (50):
\[ \frac{1}{u_2} = \frac{5}{50} - \frac{7}{50} = -\frac{2}{50} \]
Thus:
\[ u_2 = -25 \text{ m} \]

### Step 4: Calculate the distance covered by the object
The distance covered by the object (d) is:
\[ d = u_2 - u_1 = -25 - (-50) = 25 \text{ m} \]

### Step 5: Calculate the speed of the object
Speed (v) is given by:
\[ v = \frac{d}{t} \]
Where \( t = 30 \) seconds. Thus:
\[ v = \frac{25 \text{ m}}{30 \text{ s}} = \frac{5}{6} \text{ m/s} \]

### Step 6: Convert speed to km/h
To convert from m/s to km/h, we multiply by \( \frac{18}{5} \):
\[ v = \frac{5}{6} \times \frac{18}{5} = 3 \text{ km/h} \]

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

---