Question

An object is placed at a distance of $40\, cm$ from a concave mirror of focal length $15\, cm$. If the object is displaced through a distance of $20\, cm$ towards the mirror, the displacement of the image will be

• A. 36 cm towards the mirror
• B. 30 cm away from the mirror
• C. 30 cm towards the mirror
• D. 36 cm away from the mirror

Answer 1

The Correct Option is D

To solve the problem, we need to apply the mirror formula and understand the behavior of an image formed by a concave mirror.

The mirror formula is:

\frac{1}{f} = \frac{1}{v} + \frac{1}{u}

where,

• f is the focal length of the mirror. For a concave mirror, it is negative. Thus, f = -15\, cm.
• u is the object distance (always negative in mirror formula). Initially, u = -40\, cm.
• v is the image distance.

First, calculate the initial position of the image using the mirror formula:

\frac{1}{-15} = \frac{1}{v_1} - \frac{1}{40}

Simplifying, we get:

\frac{1}{v_1} = \frac{1}{-15} + \frac{1}{40} = \frac{-8 + 3}{120} = \frac{-5}{120}

v_1 = -24\, cm

This means the initial image is formed 24 cm in front of the mirror.

Now, the object is displaced 20 cm towards the mirror, making the new object distance:

u_2 = -20\, cm

Again, using the mirror formula to find the new image distance:

\frac{1}{-15} = \frac{1}{v_2} - \frac{1}{20}

Simplifying, we get:

\frac{1}{v_2} = \frac{1}{-15} + \frac{1}{20} = \frac{-4 + 3}{60} = \frac{-1}{60}

v_2 = -60\, cm

This means the new image is formed 60 cm in front of the mirror.

Thus, the displacement of the image is:

|v_2 - v_1| = |-60 - (-24)| = |-60 + 24| = 36\, cm

The image moves 36 cm away from the mirror.

Therefore, the correct answer is: 36 cm away from the mirror.