Mirror & Image Physics: Speed of Truck Image to Driver

[rishch]

Homework Statement

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

The Attempt at a Solution

Well my answer is 2 m/s but this is an example question so the solution accompanies the question and their answer is 4 m/s. The explanation they have provided is "In a plane mirror, the object and it's image always remain at the same distance from the mirror, So. when the car reverses at a speed of 2 m/s, the the image will also appear to move towards the mirror at the same speed of 2 m/s. So the speed at which the image of the truck appears to approach the car driver will be 2+2=4 m/s."

What? Everything till the last line makes sense. I tried this on paper and the image does actually move at 4 m/s relative to the truck but relative to the driver it moves only 2 m/s as the driver is moving away from the image at a rate of 2 m/s which cancels 2 m/s from the speed of the image, creating an effective speed of 2 m/s. Who's right and who's wrong?

 

[rishch]

Another way I tried to visualize this is to think of the mirror as stationary, the man a fixed distance from the mirror and the truck is moving towards the mirror. From the trucks perspective the image approaches at 4 m/s but from the stationary person's perspective the image approaches at 2 m/s.

 

[Ibix]

I agree with you. If the car starts 8m away from the truck, after 2s the truck will be 4m away, but according to the answer the image will pass through the mirror at that time.

You are also correct that an observer on the truck would see the image approaching at 4m/s. I would double check that I hadn't somehow misread the question and it is asking about the driver of the truck, then call it a day.

 

[rishch]

I copied the question verbatim so I guess they misread their own question. Especially the last line of their solution. Just double 2 m/s for absolutely no reason at all. Thanks for the clarification :)

 

[haruspex]

I suspect the question intended to ask how fast the image would seem to be moving to the driver. That is, if the driver interpreted the image as a truck approaching her from in front, at what speed would that virtual truck appear to be travelling. The answer to that would be 4m/s.

 

[rishch]

I suspect the question intended to ask how fast the image would seem to be moving to the driver. That is, if the driver interpreted the image as a truck approaching her from in front, at what speed would that virtual truck appear to be travelling. The answer to that would be 4m/s.

No it would be 2 m/s in that case as it's the same as a stationary person in front of a mirror with some other object moving towards the mirror and he's observing it. It appears to be moving at 2 m/s to the driver as his motion is in the same direction as the image. If you take the actual truck as the reference point then it moves at 4 m/s

 

[Ibix]

Hm. I think Haruspex has found an interpretation of the question that is consistent with the answer given. (S)he's arguing that the driver sees the truck coming at him at 2m/s relative to him, but knows he's backing up at 2m/s, so is able to deduce the ground-speed of the image-truck as 4m/s.

I, and apparently rishch, are interpreting "speed of approach" as speed relative to the observer being approached. The question, per Haruspex, may be using a more casual definition - for example "I was doing 40mph southbound, when I saw a car coming towards me on the wrong side of the road doing about 60mph" likely means that the badly-driven car was doing 60mph relative to the ground, not 20mph.

So - the question is (arguably) ambiguously phrased. Rishch - you've definitely understood the situation and done the maths right. Whether or not you've got the right answer depends on what quantity it is that you think the question is asking for. I'd still tend towards believing you are right - this is a physics question and some precision of language is expected.

A minor note - 4m/s is the correct answer if you replace the last "driver" in the question with "truck" or "driver of the truck". Editing fail?