# Optics question; physics homework... Answered

Hi guys, There is a new physics teacher at school and he asked us this question: an object fell into a very dark hole in the ground and you don't have a flashlight, but the sun is 30 degrees above the horizon.
Using a mirror you have to reflect light from the sun into the hole.
In one case the hole is one meter deep and in the second case it's three meters deep.

The question is: what angle should the mirror be placed in relation to the sun (or horizon) to illuminate the hole?

I don't really remember how to calculate this so if you can help me that would be great.
Thanks!

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For problems that are basically geometrical, it helps a lot to draw a picture of it.

That being said, I think it helps to draw a picture that is consistent with the assumptions of the problem.

Also regarding style, I think it will help if your drawings look kind of like the drawings in your physics book (if it's a good physics book), and less like the drawings of Pablo Picasso.

Back to the assumptions of the problem: I am naively assuming the width of the hole is much smaller than it is deep, which explains why its so dark. Also I am assuming this is a hole with vertical walls. I mean, if it were a hole that goes into the Earth at some direction other than straight down, (like the 26 degree "entrance" shaft at the pyramid of Giza) I would expect that to be stated explicity, given, as part of the problem. A hole with unspecified angle, I assume goes straight down.

Also I am assuming I can hold the mirror directly above the bottom of the hole, as long as I am careful not to drop it in the hole, and from this position I want to throw a beam of light straight into the hole; i.e. at angle parallel to the walls of the hole, and that angle is -90 degrees, from the horizon, since I'm assuming the hole goes straight down.

Assuming I can throw the light straight into the hole, i.e. the light is parallel to the walls of the hole,(and normal to the bottom of the hole), it doesn't really matter how deep the hole is. Throwing the light at angle parallel to the walls of the hole, is the best angle, independent of how deep the hole is, 1 meter, 3 meters, whatever.

RickHarris's picture is confusing to me for this reason. His picture seems to assume the mirror has to be placed above one side of the hole. Like the hole is big enough to hold a city bus, so you'd need a crane to hold the mirror above the hole. Or maybe there is fear of dropping the mirror, so the mirror must be held above the ground, on one side of the hole. Or maybe he made the picture intentionally vague, for fear of "giving away" the answer or making it too obvious.

Of course I don't care if I make the answer obvious. I try to make my explanations clear, and it is not my responsibility to make sure you actually learn the material. Seriously, that's up to you.

So yeah. Spoilers follow. Don't read further if you want to discover the answer yourself.

The sunlight comes in at angle of 30 deg above the horizon, and leaves the mirror at angle of 90 below the horizon, then it moves through an angle of 30+90=120 deg. Half of that angle change is "angle of incidence". Half is "angle of reflection". Both angle of incidence and angle of reflection are measured relative to the line normal to the plane of the mirror. Moreover, those angles have to be equal, because that's the way perfect reflection works.

So the line normal to the plane of the mirror is pointed 30 deg below the horizon. The angle of incidence is 60 deg, equal to the angle of reflection, also 60 deg. This ray of light touching he mirror moved through an angle of 60+60=120 deg.

And this is all shown, roughly the way I imagine it, in the attached picture.

Yeah, the problem was really vague, and when I heard it I began right away to try to solve it, but the teacher then commented that the problem is harder than it looks so I posted it here.

By the way, it's not homework, it's more like a tease question.

Anyway, thanks for helping me!

The angle of reflection = The angle of incidence.

I have to say you haven't told us the entire problem. You don't say what you have to find

Draw a picture.