Does the apparent length of an object change in water?

[freshman2013]

Here is a problem that showed up on my exam that I couldn't find any variation online. A stick of length L is depth D in the water. The stick is parallel to the surface of the water, and the viewer (in air) is looking down in the water right over the middle of the stick. What is the apparent LENGTH of the stick? For me, I thought that the virtual image both ends of the stick are right above the ends of the stick, so the stick would appear shallower but preserve the same length. However, the answer is wrong, and in the ray diagram from the solutions, the virtual ray that the viewer sees originates at the same depth as the stick, which is the cause of the length difference, but I don't understand at all why that would be the case. Since if draw two rays, refract them, then backtrace the refracted rays, they always appear right over the ends of the real stick. Could someone tell me what exactly is wrong with my line of thinking, that the virtual image appears right over the stick so the stick's virtual length should be unchanged?

 

[Bystander]

At what distance are you viewing the stick?

 

[freshman2013]

At what distance are you viewing the stick?

A height H above the water. It doesn't specify how high.

 

[CWatters]

For me, I thought that the virtual image both ends of the stick are right above the ends of the stick, so the stick would appear shallower but preserve the same length.

I agree with you. The ends of the image will appear almost above the ends of the stick so they will appear shallower. See diagram.

 

[CWatters]

The angle subtended by the ends at the eye will be different.. Does that make it seem longer or just nearer?

 

[freshman2013]

The angle subtended by the ends at the eye will be different.. Does that make it seem longer or just nearer?

View attachment 107330

But the exam solutions have nothing to do with what angle it subtends. It showed the virtual rays originating from the SAME depth as the stick, and taking the distance between the origins of the two rays from the ends.

 

[CWatters]

Can you post the professors diagram? It sounds wrong to me.

For info I based my first diagram on this one which shows the classic bent stick..

https://astarmathsandphysics.com/ib...t-depth-of-a-stick-in-water-html-1c7049eb.gif

 

[freshman2013]

So it's actually a fish (I used stick because it seemed easier to explain). Also, the professor said even though his solutions involved looking directly over one end of the fish, he decided to give credit to looking right above the middle of the fish (as I stated in the OP) because the question was worded very ambiguously. Still, that doesn't change the fact that I have a problem with how the rays are originating from the depth of the fish.

 

[jbriggs444]

@CWatters is making an excellent point. The diagram misses the crucial factor that the apparent depth of the fish/stick in the water will affect its apparent size.

A better diagram might assume binocular vision and use that to gauge apparent depth. One would want to trace four rays: Front end to right eye, front end to left eye, back end to right eye, back end to left eye. That would provide enough information to obtain both a depth assessment and a corresponding size assessment.

 

[A.T.]

But the exam solutions have nothing to do with what angle it subtends.

That angle defines the "apparent size" which the question asks about. Whether a correction based on "apparent distance" is included in the "apparent size" depends on the context and exact definition of "apparent size".

 

[nasu]

View attachment 107331 So it's actually a fish (I used stick because it seemed easier to explain). Also, the professor said even though his solutions involved looking directly over one end of the fish, he decided to give credit to looking right above the middle of the fish (as I stated in the OP) because the question was worded very ambiguously. Still, that doesn't change the fact that I have a problem with how the rays are originating from the depth of the fish.

The intersection of the two dotted lines at distance L' from the head of the fish has no physical meaning. The image of the tail does not form there.
So defining L' as the apparent length is flawed, irrespective of other matters mentioned already.

 

[CWatters]

+1

The image will lie somewhere along the red dotted line but to work out where you have to draw other rays such as the green one.

 

[pixel]

The diagram in post #8 is only showing one ray from the end of the fish. You need at least two rays to determine the image location. When that is done, the image will be above the fish. If the claim that the length of the image is different than the length of the fish is based on that diagram, then that is not correct.

As to its length, see my post #8 in an earlier thread on this subject: https://www.physicsforums.com/threads/apparent-depth-conceptual-question.866441/#post-5439230
I concluded there that as one moved more and more to the right, the image not only moves up but also shifts to the right. I will try to apply that to your particular case when I have time later but believe the shift was very small compared to the depth change and maybe for this problem can be neglected.

I believe the problem is asking about the image size, not its apparent size (a smaller image that is closer could appear larger than the object).

 

[CWatters]

Hopefully it's clear now that the image of the fish can/will be larger but the method the professors used to prove that is incorrect.