Question

If you look at an object at the bottom of a pool, the pool looks less deep than it actually is. a) From what you have learned, calculate how deep a pool seems to be if it is actually 4 feet deep and you look directly down on it. The index of refraction of water is \(1.33 .\) b) Would the pool look more or less deep than it actually is if you looked at it from an angle other than vertical? Answer this qualitatively, without using an equation.

Short answer

Answer: The apparent depth of the pool appears to decrease when viewed at an angle other than vertical. When viewed directly from above, the apparent depth of the pool appears to be approximately 3 feet.

Step-by-step solution

Calculate the apparent depth using Snell's Law

Snell's law relates the angle of incidence and the angle of refraction between two media with different indices of refraction. In this case, the two media are air (index of refraction \(n_1 = 1\)) and water (index of refraction \(n_2 = 1.33\)). Looking directly down into the pool means that the angle of incidence is 0 degrees. Snell's law is given by: \(n_1 \sin(\theta_1) = n_2 \sin(\theta_2)\) Since \(\sin(0) = 0\), the angle of refraction inside water (\(\theta_2\)) is also 0 degrees when viewing straight down. Thus, the path of light through the water is straight down, and we can use the relation: \(apparent\ depth\ (d') = real\ depth\ (d) \times \frac{n_1}{n_2}\) In this case, the real depth of the pool is 4 feet.

Substitute the given values and solve for apparent depth

Now, we will plug in the known values of real depth, \(n_1\), and \(n_2\) to find the apparent depth. \(d' = 4 ft \times \frac{1}{1.33}\) \(d' \approx 3 ft\) So, the pool seems to be about 3 feet deep when it is actually 4 feet deep and viewed directly from above.

Discuss the pool's depth appearance when viewed at an angle

When the pool is viewed at an angle other than vertical, the angle of incidence will be greater than 0 degrees. According to Snell's Law, as the angle of incidence increases, the angle of refraction will also increase, which means the light will bend more within the water. This bending of light coming from the bottom of the pool will cause the pool to appear shallower than it actually is, and even shallower than when viewed directly from above. So, the pool would look less deep when viewed at an angle other than vertical.