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

Answer 1

The Correct Option is C

To solve this problem, we need to understand how the image position changes for a concave mirror when the object is moved. We will use the mirror formula and calculate the positions of the image before and after the object is moved.

Step 1: Initial Image Position

When the object is 40 cm from the mirror, we use the mirror formula:

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

where \( f = -15 \) cm (since it is a concave mirror), and \( u = -40 \) cm (object distance is taken as negative in mirror convention).

Substitute these values into the formula:

\(\frac{1}{-15} = \frac{1}{v} + \frac{1}{-40}\)

Simplifying gives:

\(\frac{1}{v} = \frac{1}{-15} - \frac{1}{-40} = \frac{-40 + 15}{600} = \frac{-25}{600} = \frac{-1}{24}\)

This implies \( v = -24 \) cm.

The image is 24 cm in front of the mirror (on the same side as the object).

Step 2: New Image Position After Object Moves

The object is moved 20 cm closer to the mirror, thus, the new object distance \( u' = -20 \) cm.

Using the mirror formula again:

\(\frac{1}{-15} = \frac{1}{v'} + \frac{1}{-20}\)

Simplifying gives:

\(\frac{1}{v'} = \frac{1}{-15} - \frac{1}{-20} = \frac{-20 + 15}{300} = \frac{-5}{300} = \frac{-1}{60}\)

This implies \( v' = -60 \) cm.

The new image position is 60 cm in front of the mirror.

Step 3: Calculate Image Displacement

The initial image position was 24 cm and the new position is 60 cm. Therefore, the image displacement is:

\( |60 - 24| = 36\,\text{cm} \)

The image moves 36 cm away from the mirror.

Conclusion

Thus, the image displaces 36 cm away from the mirror, so the correct answer is 36 cm away from the mirror.