Question

The image of an object approaching a convex mirror of radius of curvature 20 m along its optical axis is observed to move from \(\frac{25}{3} \mathrm{~m}\) to \(\frac{50}{7} \mathrm{~m}\) in 30 second. The speed of the object in \(\mathrm{km} / \mathrm{hr}\) is

• A. 5
• B. 2.5
• C. 4
• D. 3

Hint:

Find the focal length from the radius, get the two object distances with the mirror formula, then divide the distance moved by the time.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
A convex mirror always forms a virtual image behind the mirror. Its focal length is \(f = \dfrac{R}{2} = 10\text{ m}\).
Use the mirror formula to find the object distance for each image distance.

Step 2: First Object Distance
For a convex mirror, taking magnitudes, \(\dfrac{1}{u} = \dfrac{1}{v} - \dfrac{1}{f}\).
For \(v = \dfrac{25}{3}\text{ m}\): \(\dfrac{1}{u} = \dfrac{3}{25} - \dfrac{1}{10} = \dfrac{1}{50}\), so \(u = 50\text{ m}\).

Step 3: Second Object Distance
For \(v = \dfrac{50}{7}\text{ m}\): \(\dfrac{1}{u} = \dfrac{7}{50} - \dfrac{1}{10} = \dfrac{1}{25}\), so \(u = 25\text{ m}\).

Step 4: Calculation of Speed
Distance moved by the object \(= 50 - 25 = 25\text{ m}\) in \(30\text{ s}\).
\(\text{Speed} = \dfrac{25}{30} = \dfrac{5}{6}\text{ m/s} = \dfrac{5}{6}\times 3.6 = 3\text{ km/hr}\).

Final Answer:
The speed of the object is 3 km/hr. \[ \boxed{\text{(D) }3} \]