A 4.00-m -long pole stands vertically in a lake having a...

A $4.00-\mathrm{m}$ -long pole stands vertically in a lake having a depth of 2.00 $\mathrm{m}$ . The Sun is $40.0^{\circ}$ above the horizontal. Determine the length of the pole's shadow on the bottom of the lake. Take the index of refraction for water to be $1.33 .$

Key Concepts

Geometric Analysis of Light Paths

This concept involves using geometry and trigonometry to trace the path of light as it travels through different media. In problems involving shadows or projections (such as a shadow cast on a submerged surface), geometric analysis is used to relate the angles determined by Snell’s Law to distances and positions on the surfaces involved.

Snell's Law

Snell's Law is the fundamental relationship that describes how light is refracted at the boundary between two media with differing indices of refraction. It relates the sines of the angles of incidence and refraction to the indices of refraction of the two media, allowing one to calculate the direction of the light ray after crossing the interface.

Index of Refraction

This concept refers to a quantitative measure of how much light slows down when it passes into a medium compared to its speed in a vacuum. It is a key parameter that determines how light bends (or refracts) at the interface between two media with different refractive indices.

Transcript

00:01 So here we have a four meters tall pole, four meters tall, standing vertically inside of a lake.

00:09 So here's our lake.

00:13 And the lake has a depth of two meters.

00:16 So this is two meters down.

00:19 Determine the length of the pole's shadow at the bottom of the lake.

00:24 They also give us that the sun, put the sun over here, is 40 degrees above the horizontal here.

00:32 And they say that's 40 degrees above the horizontal.

00:38 It's also just go from the top.

00:41 This will also be 40 because the sun is extremely far away.

00:46 They said the index of a fraction of water is equal to 1 .3.

00:50 We'll assume the index of the air is 1 .00.

00:53 And the goal is to figure out this shadow.

00:57 So i'm going to just assume that this shadow will find the top of the shadow and the bottom of the shadow, and that'll get us the length of the shadow.

01:06 In the water.

01:07 So i'm gonna draw the refracted rays somewhat like this, and we just are trying to figure out what is the length there of that shadow, let's call it now.

01:20 And so here we've got some rays to work with.

01:26 And actually i mistrewed this one, because first it does go through the air.

01:38 Let's drop in blue.

01:40 And then it refracts back down into the water.

01:45 And so we've got a couple lengths and distances to work with, where l is actually this l.

01:50 All right.

01:53 All right, so let's use some triangles.

01:54 I know this angle here is 40 degrees.

01:57 I know this is a right angle, assuming that pole is just perfectly there.

02:03 What i would love to do is find some other angles, namely these two, which should be the same because it's all symmetrical.

02:13 In terms of snell's law.

02:15 So let's find that theta first.