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Anderson Video - Wall Mirror Question

Professor Anderson
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 Let's say we're standing in front of the wall mirror. And the wall mirror is not half your height. It is maybe one quarter of your height. Okay, so here's the wall mirror. And here you are. Okay, and let's, let's move this mirror up just a little bit so it's even with the head of the person. Okay, and the question is the following. If you are height H and this thing is H over 4 how far back do you have to stand to see your shoes? Okay, namely you're standing next to this mirror and you can't quite see your shoes. Okay, how far back do you have to go in order to see your shoes? And this thing is wall-mounted. Okay, it's flat. This is our wall. It's a flat mirror, it's perfectly vertical and you're just going to walk back on a level ground. How far back do you have to go to see your shoes? A lot of people said, "oh, it's got to be 2h." Some people said, "well it's 1/4 the size of the mirror so it's got to be 4H." Let's see if we can think about how this is going to work. What we said was that rays that bounce off a mirror bounce off the exact same angle they came in at. So, if we think about the bottom of this mirror and a ray going from the bottom of the mirror to the person's eye where did it come from? It had to come from about there. Okay, incoming angle has to equal reflected angle. All right, so at this distance right now, you can see maybe your midsection, right, your bellybutton. But now let's take this person and let's move them far away and see what they can see. Okay, let's extend our floor a little bit. And now that person is right here. The light ray that came to their eye had to come at an exact same angle. Okay, can they see their shoes? No, they see their bellybutton again. As the person steps back those rays start to narrow down. So, even though you've increased the distance that the ray is traveling the angle has also decreased. And so when can you see your shoes? Never. You can never see your shoes. If everything is flat and vertical and horizontal floor you can never see your shoes, which is kind of weird to think about, right? If I'm standing and looking in the mirror surely I must be able to back up enough to see my shoes. That's not the case. Okay, you'll always see the exact same point on your body, no matter how far back you go.