How to calculate horizontal distance?

Question

A rock lies on the bottom of a 10m deep lake. You are in a boat directly above the rock, and you can see it well. As the boat moves away from the rock, after a certain horizontal distance x. you cannot see the rock anymore. What is the distance? The refractive distance of water is 1.33

Answer 1

Snell's Law of Refraction:

 

n_wsinθ_w = n_asinθ_a

 

Total internal reflection occurs when θ_a = 90°, so sinθ_a = 1.

 

n_wsinθ_w = n_asinθ_a

n_wsinθ_w = n_a·1

sinθ_w = n_a/n_w

θ_w = sin^-1(n_a/n_w) = sin^-1(1/1.33)

 

Use your calculator to compute θ_w.  If you draw a diagram, you'll see that:

 

tan(θ_w) = d/10

 

Where 10 = the depth of the lake and d = your horizontal distance from the rock.  Solve for d (in meters).