A submarine is 300 m horizontally from the shore of a...

A submarine is $300 \mathrm{~m}$ horizontally from the shore of a freshwater lake and $100 \mathrm{~m}$ beneath the surface of the water. A laser beam is sent from the submarine so that the beam strikes the surface of the water $210 \mathrm{~m}$ from the shore. A building stands on the shore, and the laser beam hits a target at the top of the building. The goal is to find the height of the target above sea level. (a) Draw a diagram of the situation, identifying the two triangles that are important in finding the solution. (b) Find the angle of incidence of the beam striking the water-air interface. (c) Find the angle of refraction. (d) What angle does the refracted beam make with the horizontal? (e) Find the height of the target above sea level.

Key Concepts

Snell's Law

Snell's Law is the fundamental principle that describes how light bends when passing from one medium into another with a different index of refraction. It relates the sine of the angle of incidence to the sine of the angle of refraction by the ratio of the speeds of light (or equivalently the indices of refraction) in the two media. This principle is key in determining the behavior of the light beam at the interface between water and air.

Refraction at an Interface

Refraction refers to the change in direction of a light ray when it passes from one medium to another due to a change in its propagation speed. Understanding how the path of a ray deviates at the boundary between two media is critical for calculating both the angle of incidence as well as the angle of refraction, which then influences the subsequent trajectory of the beam.

Right Triangle Geometry

Right triangle geometry is essential when analyzing the path of the light beam segmented into underwater and above-water portions. By modeling each segment as a right triangle, one can make use of trigonometric ratios to find unknown distances and angles. This approach simplifies the complex path of the beam into a series of manageable calculations.

Trigonometric Relationships

Trigonometric relationships, including functions such as sine, cosine, and tangent, allow for the determination of angles and lengths in the diagrams constructed from the problem. These relationships are crucial for calculating the angles of incidence, refraction, and the overall trajectory of the light beam by relating side lengths of the triangles to the angles within them.

Transcript

00:01 Consider the case of a submarine some distance from shore.

00:04 First we'll draw the shore.

00:10 Here is the lake that our submarine is in.

00:16 Now we'll sketch our submarine.

00:30 The submarine is at a depth of 100 meters.

00:40 Additionally, it is 300 meters from shore.

00:54 The submarine sends out a laser beam that strikes the surface of the water 210 meters from shore.

01:17 That means that this distance, is 90 meters.

01:30 Because we know that this distance is 90 meters and the vertical distance is 100 meters, we can use sokatoa to solve for the missing angle.

01:40 So let me write sokatoa, where this represents sign being a function of the opposite side and the hypotenuse, cosine being a function of the adjacent side and the hypotenuse, and tangent being a function of the opposite side and the adjacent side.

02:02 So we only have the adjacent side, which is 100 meters, and the opposite side, which is 90 meters.

02:09 So we want to use tangent to solve for this angle.

02:20 To solve for theta, we need to take the inverse tangent.

02:35 So opposite is 90 meters.

02:38 Adjacent is 100 meters.

02:42 So we find theta is the inverse tangent of 0 .9.

02:50 This is equal to approximately 42 degrees of theta equals 42.

03:04 Degrees and that's this angle.

03:11 Now when this laser beam strikes the surface of the water, it's going to refract so we actually need the angle in terms of the angle relative to the vertical shown right here.

03:24 Whoops, that's not a very good vertical line.

03:33 When the light refracts, it will have a different angle.

03:44 Due to geometry, we know that this red angle theta is equal to this angle...

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