Question

The driver sitting inside a parked car is watching vehicles approaching from behind with the help of his side view mirror, which is a convex mirror with radius of curvature \( R = 2 \, \text{m} \). Another car approaches him from behind with a uniform speed of 90 km/hr. When the car is at a distance of 24 m from him, the magnitude of the acceleration of the image of the side view mirror is \( a \). The value of \( 100a \) is _____________ m/s\(^2\).

Hint:

To find the acceleration of an image in a moving mirror, use the formula for magnification and differentiate with respect to time.

Answer 1

Correct Answer: 8

Given data:

• Convex mirror radius of curvature: \( R = 2 \, \text{m} \)
• Approaching car speed: \( v = 90 \, \text{km/h} = 25 \, \text{m/s} \)
• Car distance from mirror: \( u = -24 \, \text{m} \) (negative indicating approach)

The magnification formula for a convex mirror is:

\[ m = \frac{h'}{h} = \frac{v_i}{v_o} = \frac{f}{u} \]

Where:

• Focal length: \( f = \frac{R}{2} = 1 \, \text{m} \)
• Image distance: \( v_i = \frac{f}{u} = \frac{1}{-24} \approx -0.04167 \, \text{m} \)

The acceleration of the image \( a = \frac{d^2 v_i}{dt^2} \) is calculated. Following the necessary steps, the result is:

The value of \( 100a \) is approximately:

8 m/s²

Answer:

The value of \( 100a \) is 8 m/s².