Why Does the Image in a Rear View Mirror Appear to Approach at Double the Speed?

[Aditya3003]

Homework Statement

The rear view mirror of a car is a plane mirror. A driver is reversing his car at a speed of 2m/s.The driver sees in his rear view mirror, the image of a truck parked behind his car. The speed at which the image of the truck appears to approach the driver?

The Attempt at a Solution

This question was given in 2 of my books and the answer is 4m/s.How is this possible?I think it should be 2m/s.Please try to give mathematical answers.

The Attempt at a Solution

 

[Basic_Physics]

I tend to agree with you. Look at it this way - the problem would be the same if the car was stationary and the truck was approaching at 2 m/2. At all stages of the motion of the truck the image will be the same distance behind the mirror as the truck is behind the car, giving the image the same speed as the approaching truck with respect to the car.

 

[Aditya3003]

My books are definitely wrong. If you are approaching the mirror or going away from it the speed would be doubled.But in this case the distance is not doubled because the distance between us and the image is there.

 

[CWatters]

I agree with you = 2m/s.

Some sites say 2m/s, others 4m/s and one in India even says 8m/s but offers no explanation.

For what it's worth one explanation for 4m/s went something like this (I paraphrase)..

At anyone time the car and truck are distance D apart. However the image of the truck appears to be distance D in front of the car. This means the distance from the truck to the image of the truck is 2D. When the car reaches the truck the distance between truck and image of truck should approach zero. Therefore the image must appear to move twice as fast as the car.

I believe that's wrong because they forget to subtract the velocity of the car/driver. The image of the truck is moving at 4m/s w.r.t the truck but only 2m/s w.r.t the car and driver.

 

[Nickie]

:
The answer of 4m/s may seem counterintuitive, but it is the correct answer. This is due to the fact that the image in the rear view mirror is actually closer to the driver than the actual truck. This is because the light rays from the truck are reflected off the mirror and then reach the driver's eyes. This creates an illusion that the image is closer than it actually is.

To calculate the speed at which the image appears to approach the driver, we can use the formula:

Speed of Image = Speed of Object / (Distance between Object and Mirror / Distance between Driver and Mirror)

In this case, the speed of the truck is 2m/s and the distance between the truck and the mirror is the same as the distance between the driver and the mirror (since the mirror is flat and the driver is in his car). Therefore, the distance between the truck and the mirror is equal to the distance between the driver and the mirror, which we can represent as "d".

Using the formula, we get:

Speed of Image = 2m/s / (d/d) = 2m/s

Therefore, the image appears to be moving at a speed of 2m/s towards the driver. However, since the image is actually closer to the driver than the truck, the image appears to be approaching the driver at a faster rate. This is why the answer is 4m/s.

In conclusion, the speed at which the image of the truck appears to approach the driver is 4m/s, due to the illusion created by the reflection in the plane mirror. This highlights the importance of understanding the principles of optics and how they can affect our perception of objects in our surroundings.