Minimum Height of a Vertical Reflector

[Const@ntine]

Homework Statement

Find the minimum height of a vertical flat reflector, in which a 1,78 m person can see its full reflection.

Homework Equations

M = h'/h

The Attempt at a Solution

I really don't know what to do here, and I feel as if I'm missing some crucial element. The chapter about vertical reflectors is very small (3 pages or so) and really has no other exercises besides a theoretical one based on this design:

I tried going at it by creating "formulas" with the different angles, but nothing really worked. I assume the hidden detail is in that the person can see its full reflection, but other than h = h', I'm not saying anything else. It might be an issue in translation as well.

Any help is appreciated!

 

[haruspex]

I do not understand the diagram.
My interpretation of the question is that you are to mount a mirror vertically on the wall such that a 1,78m person standing in front of it will see their whole reflection. The length of the mirror is to be minimised.
That does not look at all like your diagram.

 

[Const@ntine]

I do not understand the diagram.
My interpretation of the question is that you are to mount a mirror vertically on the wall such that a 1,78m person standing in front of it will see their whole reflection. The length of the mirror is to be minimised.
That does not look at all like your diagram.

Well, in that particular "chapter" (O2.1 of Serway's Physics, 8th Ed, Vol 2) that's the only instance it deals with. The person is on the far left, and the reflector/mirror is vertical to the floor (with the reflection on the far right). Pretty much this:

 

[haruspex]

Well, in that particular "chapter" (O2.1 of Serway's Physics, 8th Ed, Vol 2) that's the only instance it deals with. The person is on the far left, and the reflector/mirror is vertical to the floor (with the reflection on the far right). Pretty much this:

View attachment 203513

Ok, so draw a side view showing the light paths to the eye from the extremes of the body.

 

[Const@ntine]

Ok, so draw a side view showing the light paths to the eye from the extremes of the body.

That's what I have on my first post is supposed to be. I did it according to the book's orders: One ray of light is perpendicular to the reflector, and parallel to the ground. It hits the reflective surface, and returns back to the person, following the exact same path. The other ray of light follows a diagonal path, and is reflected by the surface, thus creating the prolapse/reflection angles. If we "continue" the rays, they meet at some point, and thus we can "draw" the (imaginary) reflection with height h'.

 

[haruspex]

One ray of light is perpendicular to the reflector,

My eyes are not quite at the top of my head.

The other ray of light follows a diagonal path, and is reflected by the surface

Neither do my eyes emit light. The light comes from what I am looking at. If I am looking at my feet in the mirror, what path do the rays take?

 

[Const@ntine]

My eyes are not quite at the top of my head.

Neither do my eyes emit light. The light comes from what I am looking at. If I am looking at my feet in the mirror, what path do the rays take?

Yeah, I know it's not exactly a model that represents the physical world, but that's the only example/exercise the book has. You are supposed to act as if the object/person/source of light are one and the same.

What I'm guessing you mean is something like this, right?

 

[ehild]

That's what I have on my first post is supposed to be. I did it according to the book's orders: One ray of light is perpendicular to the reflector, and parallel to the ground. It hits the reflective surface, and returns back to the person, following the exact same path. The other ray of light follows a diagonal path, and is reflected by the surface, thus creating the prolapse/reflection angles.

What do you know about the prolapse/reflection angles?
You need to see your eye and your feet with the mirror. You see the eye with the light ray emerging from your eye, going perpendicularly to the mirror and reflecting from it, then returning to your eye.
The right emerging from your foot reaches the mirror with oblique incidence. it is reflected, and the reflected light reaches your eye. What is the relation between the angles the incident and reflected rays make with the horizontal?

 

[Const@ntine]

What do you know about the prolapse/reflection angles?
You need to see your eye and your feet with the mirror. You see the eye with the light ray emerging from your eye, going perpendicularly to the mirror and reflecting from it, then returning to your eye.
The right emerging from your foot reaches the mirror with oblique incidence. it is reflected, and the reflected light reaches your eye. What is the relation between the angles the incident and reflected rays make with the horizontal?

For this particular problem? Nothing. All I get is the person's height, and the "it is able to see its full reflection". I tried to go at it with angles, sines and whatnot, but since the only known quantity is the height, I'm not able to reach a conclusion.

 

[ehild]

For this particular problem? Nothing. All I get is the person's height, and the "it is able to see its full reflection". I tried to go at it with angles, sines and whatnot, but since the only known quantity is the height, I'm not able to reach a conclusion.

You need to know the laws of reflection. Read http://www.physicsclassroom.com/class/refln/Lesson-1/The-Law-of-Reflection

 

[Const@ntine]

You need to know the laws of reflection. Read http://www.physicsclassroom.com/class/refln/Lesson-1/The-Law-of-Reflection

EDIT: Hmm, turns out those things in the link are much further down in my book. I'll have to go through them and check back.

 

[haruspex]

EDIT: Hmm, turns out those things in the link are much further down in my book. I'll have to go through them and check back.

That is the way I was trying to lead you along, but if you don't know the law of reflection then there is another way.
Go back to your original diagram, showing the image of the person behind the mirror. You have correctly shown the image to be as far behind the mirror as the viewer is in front. Now you can forget the reflections and just think about the viewer and image. Draw the light rays from image feet to viewer eye and from image scalp to viewer eye. Where do they pass through the mirror?

 

[BenjaminLovesQM]

It should be half his height

 

[BenjaminLovesQM]

Since the angle of reflection is the same as the angle of incidence, two congruent triangles can be formed when u draw the diagram properly (yours isn't correct). Then, for him to see the top of his head, a ray of light from the tip of his head has to reach his eyes. Therefore, another two congruent triangles can be formed. With two pairs of congruent triangles, the height of the vertical reflector should be half his height.

 

[Const@ntine]

Okay, I'm going to be honest and say that I'm still not getting it. Geometry was never my strong suit, so all this triangle talk is leaving me rather dumbfound. And I really don't have the time to brush up on geometry (less than a month till the finals and I haven't finished even one subject), so could someone draw the correct diagram? I've been at it for a whole day now, so I think it's time to give up and continue to the rest.

Thanks a ton for the help so far, but I'm really not getting it. Not trying to sound ungrateful, but if I'm not getting it by now, I probably never will, and I'd like to see the correct version in case I come across something similar later.

 

[haruspex]

Okay, I'm going to be honest and say that I'm still not getting it. Geometry was never my strong suit, so all this triangle talk is leaving me rather dumbfound. And I really don't have the time to brush up on geometry (less than a month till the finals and I haven't finished even one subject), so could someone draw the correct diagram? I've been at it for a whole day now, so I think it's time to give up and continue to the rest.

Thanks a ton for the help so far, but I'm really not getting it. Not trying to sound ungrateful, but if I'm not getting it by now, I probably never will, and I'd like to see the correct version in case I come across something similar later.

Please verify that you tried what I wrote in the last sentence of post #12.

 

[Const@ntine]

Please verify that you tried what I wrote in the last sentence of post #12.

Yeah, I created that triangle (person's eye to reflection's scalp & reflection's feet to person's eye) but I'm just not getting how to use the data to come to a conclusion. I'd greatly appreciate the finished sketch to see where I'm going wrong and what I should think of. I'm very thankful of all the help thus far, but I'm stuck on this particular one.

 

[haruspex]

Yeah, I created that triangle (person's eye to reflection's scalp & reflection's feet to person's eye) but I'm just not getting how to use the data to come to a conclusion.

Ok.
Note where those lines cross the mirror. That shows which parts of the mirror are being used in those reflections. Reflections from the rest of the body will cross between those points.
In relation to the height of the viewer's eye, how high is the lower crossing point?

 

[Const@ntine]

Ok.
Note where those lines cross the mirror. That shows which parts of the mirror are being used in those reflections. Reflections from the rest of the body will cross between those points.
In relation to the height of the viewer's eye, how high is the lower crossing point?

The lowest point is at the level of half his heigh, give or take. The highest point is pretty much at his scalp.

 

[haruspex]

The lowest point is at the level of half his heigh, give or take.

No give or take, exactly half the height of the eyes.

The highest point is pretty much at his scalp.

If you draw a horizontal line through the eyes, the situation above it is completely analogous to that below it. Just turn it upside down and see where the scalp-to-eye reflection point is in relation to the scalp and the eye: half way between again.
What does that tell you about how the distance between the two extreme reflection points relates to the person's height?

 

[Const@ntine]

No give or take, exactly half the height of the eyes.

If you draw a horizontal line through the eyes, the situation above it is completely analogous to that below it. Just turn it upside down and see where the scalp-to-eye reflection point is in relation to the scalp and the eye: half way between again.
What does that tell you about how the distance between the two extreme reflection points relates to the person's height?

Well... I'm still not getting it. What does "the height of the eyes" have to do with it? I'm really not getting what this drawing is supposed to represent. If it's not too much trouble could I just get a complete drawing? Because I don't think I'll ever get it through my head just with words. I'm having trouble visualizing this one.

 

[haruspex]

Well... I'm still not getting it. What does "the height of the eyes" have to do with it? I'm really not getting what this drawing is supposed to represent. If it's not too much trouble could I just get a complete drawing? Because I don't think I'll ever get it through my head just with words. I'm having trouble visualizing this one.

It's not a question of my trouble. This is simply not how this should work. It's your homework; you draw the diagram according to what I described and post it. If it's wrong, I will tell you in what way it is wrong and why.

 

[Const@ntine]

It's not a question of my trouble. This is simply not how this should work. It's your homework; you draw the diagram according to what I described and post it. If it's wrong, I will tell you in what way it is wrong and why.

Well, in that case, I guess this is the diagram:

 

[haruspex]

Well, in that case, I guess this is the diagram:

View attachment 203604

Great.
Add one more line, the horizontal joining the two sets of eyes.
Define some points on the mirror vertical:
A at ground level, B where the lower ray crosses, C at eye level, D where the upper ray crosses, E at scalp level.

What is the relationship between the following pairs of lengths:
AB and BC
CD and DE
AE and h, the height of the viewer
AE and BD?
Which length is the minimum length of the mirror?

 

[Const@ntine]

Great.
Add one more line, the horizontal joining the two sets of eyes.
Define some points on the mirror vertical:
A at ground level, B where the lower ray crosses, C at eye level, D where the upper ray crosses, E at scalp level.

What is the relationship between the following pairs of lengths:
AB and BC
CD and DE
AE and h, the height of the viewer
AE and BD?
Which length is the minimum length of the mirror?

Updated sketch:

In this case, AE is the height, h. BD is the minimum height of the mirror, h_min, which is what I'm trying to find. As for the rest, I'm not seeing the connections between them. They probably have something to do with the similar triangles, but like I said, geometry is really not my strong suit, so I'm at a loss.

PS: Let me check it again when I'm back from class in about 6 hours or so.

 

[Const@ntine]

Well, I think I finally did it.

I looked at the two triangles above the eye-to-eye horizontal, and go with SAS, I found out they're congruent, so CD = DE. Same with the ones ellow the horizontal, so AB = BC. So:

AE = AB + BC + CD + DE = 2BC + 2CD = 2BD <=> h_min = h/2

I'm not sure if that's entirely correct, but this is purely a geometry issue. Either way, thanks a ton for the help and patience everyone!

 

[haruspex]

not sure if that's entirely correct,

It is.