Question

A plane mirror is placed in \(y-z\) plane facing towards negative \(x\)-axis. The mirror is moving parallel to \(y\)-axis with a speed of 2 \(\mathrm{cm} / \mathrm{s}\). A point object \(\mathrm{p}\) move is moving in front of the mirror with a velocity \((1 \mathrm{~cm} / \mathrm{s})\) \(\hat{i}+(1 \mathrm{~cm} / \mathrm{s}) \hat{j}\). Then which of the following statements is incorrect? (1) The velocity of image is \((-\hat{i}+\hat{j}) \mathrm{cm} / \mathrm{s}\) (2) The velocity of image with respect to mirror is \(-(\hat{i}+\hat{j}) \mathrm{cm} / \mathrm{s}\) (3) The velocity of image with respect to object is \(-(2 \hat{i}) \mathrm{cm} / \mathrm{s}\) (4) The unit vector in the direction of reflected ray is \(\frac{(\hat{i}+\hat{j})}{\sqrt{2}}\)

Short answer

Statement (1) is incorrect.

Step-by-step solution

- Understand the velocities

The point object p is moving with a velocity of \((1 \, \mathrm{cm/s}) \hat{i} + (1 \, \mathrm{cm/s}) \hat{j}\). The mirror is moving with velocity \(2 \, \mathrm{cm/s} \hat{j}\).

- Determine the velocity of the image

Since the mirror is moving in the \( \hat{j} \)-direction, the velocities of the object and the mirror should be treated separately in that direction.For the x-component, the image's velocity will be the negative of the object's velocity component, so it will be \(-(1 \hat{i}) = -\hat{i}) \mathrm{cm/s}\).For the y-component, considering both the object's motion and the mirror's motion, the velocity of the image with respect to the ground is \( (1 \hat{j} + 2 \hat{j} = 3 \hat{j}) \mathrm{cm/s}\).So, the total velocity of the image is \(-\hat{i} + 3\hat{j}\).

- Analyze the given statements

Now let's revisit the given options in light of our calculations:1. The velocity of image is \(( - \hat{i} + \hat{j} ) \mathrm{cm/s} \). This is incorrect, the correct velocity should be \(( - \hat{i} + 3 \hat{j} ) \).2. The velocity of the image with respect to the mirror is obtained by subtracting the velocity of the mirror from the image's velocity in the y-direction: \(( - \hat{i} + 3 \hat{j} - 2 \hat{j} = - \hat{i} + \hat{j} ) \).3. The velocity of the image with respect to the object is obtained by subtracting the velocity of the object from the image's velocity: \(( - \hat{i} + 3 \hat{j} - \hat{i} - \hat{j} ) \), which simplifies to \(-2 \hat{i} + 2 \hat{j}\).4. The unit vector in the direction of the reflected ray's velocity remains unchanged because it depends on direction and not on the exact speed.

- Identify the incorrect statement

From the analysis:- Statement 1 is incorrect as it does not match the actual velocity calculated.- All other statements are correct on reviewing.

Key concepts

Velocity of Image

When dealing with mirrors, we often need to find the velocity of the image created by the mirror. Think of plane mirrors as making a virtual image on the opposite side of the mirror, at an equal distance from it as the real object. The image follows a simple rule for its velocity: its velocity in the direction perpendicular to the mirror (in our case, the x-direction) is the negative of the object’s velocity, while in the direction parallel to the mirror (y-direction), it is just the sum of the object and mirror's velocities.

So if a point object moves with velocity \( (1 \, \text{cm/s}) \hat{i} + (1 \, \text{cm/s}) \hat{j} \) and the mirror moves with velocity \( 2 \, \text{cm/s} \hat{j} \), the image's velocity would be calculated as follows:
* For x-component: \( - (1 \, \text{cm/s}) \hat{i} \)
* For y-component: \(1 \, \text{cm/s} \hat{j} + 2 \, \text{cm/s} \hat{j} = 3 \, \text{cm/s} \hat{j} \)
Thus, the velocity of the image is \( -\hat{i} + 3\hat{j} \). This understanding is key in approaching problems involving the motion of images in mirrors.

Relative Velocity

Relative velocity is the velocity of one object as observed from another moving object. In problems involving mirrors, it's essential to consider how the mirror’s motion influences the perceived velocity. To find the relative velocity of the image with respect to the mirror, subtract the mirror's velocity from the image's velocity.

Using the previous values, the image's velocity was \( - \hat{i} + 3 \hat{j} \), and the mirror's velocity was \( 2 \, \text{cm/s} \hat{j} \). The relative velocity of the image with respect to the mirror is:
\( (-\hat{i} + 3, \text{cm/s} \hat{j} ) - ( 2 \, \text{cm/s} \hat{j} ) = -\hat{i} + \hat{j} \).
Understanding relative velocity helps pinpoint the actual movement of the image in comparison to another motion, making it clear how objects interact in dynamic environments.

Reflection in a Mirror

Reflection in a plane mirror follows some very straightforward rules. A plane mirror creates a virtual image that is a mirror image of the object across the mirror plane. This virtual image is at the same distance behind the mirror as the object is in front of it. The mirrored image flips the components of the velocity in the direction perpendicular to the mirror (effectively reversing it).

Given a mirror moving parallel to its surface, we consider both the object's movement and the mirror's. This is crucial for calculating how the reflected image moves. The object's velocities in x and y directions remain a significant factor, while the mirror's velocity affects the y-direction motion of the image.

For example, with our object having velocities \( (1 \, \text{cm/s}) \hat{i} + (1 \, \text{cm/s}) \hat{j} \) and the mirror moving at \( 2 \, \text{cm/s} \hat{j} \), the reflected image's motion would combine these velocities accordingly. This visually intuitive principle helps solve mirror-related problems efficiently.

Coordinate System Analysis

Using a coordinate system, specifically placing the mirror in the \(y-z\) plane and observing the velocities in terms of \( \hat{i} \) and \( \hat{j} \, \hat{k} \), simplifies the determination of velocity components and their interactions. The coordinates help determine how each component contributes to the overall motion.

Analyzing the motion of an object and its image through systems involves decomposing velocities into components and observing their alignment with the coordinate axes. For instance, the object with velocity \( (1 \, \text{cm/s}) \hat{i} + (1 \, \text{cm/s}) \hat{j} ) \) and the mirror moving at \( 2 \, \text{cm/s} \hat{j} \), treated additionally through the coordinate system, help us maintain clarity in calculating the image's velocity.
Using coordinates allows systematic breakdown, ensuring no step is overlooked and every velocity component is appropriately accounted for. This method is especially useful in understanding movement dynamics involving mirrors and reflections.