Question

You are looking down into a swimming pool at an angle of \(20^{\circ}\) relative to the vertical, and you see a coin at the bottom of the pool. This coin appears to you to be at a) a lesser depth than it really is. b) the same depth as it really is. c) a greater depth than it really is.

Short answer

Answer: The coin appears to be at a shallower depth than its actual depth.

Step-by-step solution

Understand Snell's Law of Refraction

Snell's Law of Refraction states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the refractive indices of the two materials involved. Mathematically, it can be represented as: sin(θ1) / sin(θ2) = n2 / n1 Where: θ1 = angle of incidence θ2 = angle of refraction n1 = refractive index of the first medium (water) n2 = refractive index of the second medium (air)

Apply Snell's Law on the problem

We are given the angle the viewer is looking down into the pool, which is 20° relative to the vertical. So, we have the angle of incidence (θ1) as 70° (90° - 20°). We need to find the angle of refraction (θ2). The refractive index of air (n2) is approximately 1, and the refractive index of water (n1) is approximately 1.33. Using Snell's Law: sin(70°) / sin(θ2) = 1 / 1.33

Calculate the angle of refraction (θ2)

To calculate the angle of refraction (θ2), we can rearrange the formula obtained in the previous step: sin(θ2) = sin(70°) * 1.33 / 1 θ2 = arcsin(sin(70°) * 1.33) θ2 ≈ 48.6°

Analyze the change in the apparent depth

Since the angle of refraction (48.6°) is less than the angle of incidence (70°), light is bending away from the normal (the vertical line). This makes the coin at the bottom of the pool appear closer to the surface, hence the observed depth looks shallower than the actual depth. Therefore, the correct answer is: a) the coin appears to be at a lesser depth than it really is.