Problem 24 - Master Resource Book in JEE Main Physics

Question

A plane mirror is approaching you at \(10 \mathrm{cms}^{-1}\). Your image shall approach you with a speed of (a) \(+10 \mathrm{cms}^{-1}\) (b) \(-10 \mathrm{cms}^{-1}\) (c) \(+20 \mathrm{cms}^{-1}\) (d) \(-20 \mathrm{cms}^{-1}\)

Step-by-Step Solution

Answer

The image approaches at \(-20 \text{ cms}^{-1}\), option (d).

Step 1: Understanding the mirror-image relationship

In a plane mirror, the image is as far behind the mirror as the object is in front of it. As the mirror moves, imagine the image moving twice the displacement compared to the mirror speed.

Step 2: Calculate the image speed relative to the mirror

If the mirror moves towards the object at a speed of \(10 \text{ cm/s}\), the image moves towards the mirror at the same speed but in the opposite direction relative to the mirror.

Step 3: Calculate the absolute speed of the image towards the object

Since both the image and the mirror move towards the object, the speed at which the image approaches the object is the sum of their individual speeds, which is \(10 + 10 = 20 \text{ cm/s}\).

Step 4: Identify the direction and correct answer

The image approaches the object at \(-20 \text{ cm/s}\) (negative indicating the image is moving towards you, as direction is considered towards the object), which matches option (d).

Key Concepts

Image Formation

In the context of plane mirrors, understanding image formation is crucial. Plane mirrors create a mirrored image that appears to stand behind the mirror at the same distance from the mirror as the object in front of it. This is because light reflects off the mirror surface, causing your brain to interpret the light paths as originating from behind the mirror.
When the object moves, its mirrored image will also move. If the object moves closer to the mirror, the image seems to move towards the mirror as well. Likewise, if the object moves away from the mirror, the image will seem to move away from the mirror.
Some key features of image formation in plane mirrors include:

• The image is a virtual image, meaning it cannot be projected onto a screen.
• The image is laterally inverted, meaning left and right are swapped.
• The size of the image is identical to the size of the object.
• The distance from the image to the mirror is equal to the distance from the object to the mirror.

Relative Motion

Relative motion describes the motion of an object as observed from another moving reference point. When dealing with mirrors, it's important to consider the relative motion between the mirror and the object.
In the case of the plane mirror moving towards you at a speed of 10 cm/s, you must consider how the image behind the mirror is moving relative to both you and the mirror. Since the mirror and image both approach you, the mirror acts as a point of reference.
Key considerations include:

• The concept of relative motion requires two objects or reference points to compare movement.
• The image will move at the same speed as the mirror, relative to the mirror, but in the opposite direction.
• This means if the mirror is approaching you, the image is also effectively moving towards the mirror.

Speed Calculation

The speed calculation for a moving image in a plane mirror is slightly counterintuitive but vital to solve problems correctly. Let's work through the steps:
When the mirror moves towards you at 10 cm/s, the image moves towards the mirror at the same speed relative to the mirror (thus effectively towards you). The relative speed between you and the image is calculated as the sum of the speeds of both the mirror and the image relative to the mirror.
Hence, given the mirror's speed of 10 cm/s, the image approaches you at:

• The mirror's speed of 10 cm/s toward the object.
• The image's additional speed towards the mirror, also 10 cm/s.

Adding these, the image approaches you at a combined speed of 20 cm/s. This is directed towards you, hence considered negative in this context, explaining the answer of (-20 cm/s).
By understanding these essential details, you grasp how quick calculations achieve accurate results, especially when considering relative motion.