Problem 28 - College Physics

Question

\(\cdot\) To a person swimming 0.80 \(\mathrm{m}\) beneath the surface of the water in a swimming pool, the diving board directly overhead appears to be a height of 5.20 \(\mathrm{m}\) above the swimmer. What is the actual height of the diving board above the surface of the water?

Step-by-Step Solution

Answer

The actual height of the diving board above the water is approximately 3.11 meters.

Step 1: Understanding Snell's Law

The apparent depth of an object submerged in water is different from its actual depth when viewed from above due to refraction. The relationship between apparent depth and actual depth can be determined by Snell's Law, which relates the indices of refraction of two media.

Step 2: Applying Apparent Depth Formula

Use the apparent depth formula: \( d_{apparent} = d_{actual} \times \frac{n_{water}}{n_{air}} \). Given that \( d_{apparent} = 5.20 \text{ m} \) and \( n_{water} = 1.33 \), \( n_{air} = 1.00 \), solve for \( d_{actual} \).

Step 3: Calculating Actual Depth

Rearrange and solve \( d_{actual} = \frac{d_{apparent} \times n_{air}}{n_{water}} \). Substitute the values to get \( d_{actual} = \frac{5.20 \times 1.00}{1.33} \approx 3.91 \text{ m} \).

Step 4: Determine Height Above Water

To find the height of the diving board above the water surface, subtract the swimmer's depth from the actual depth: \( 3.91 \text{ m} - 0.80 \text{ m} = 3.11 \text{ m} \).

Key Concepts

Snell's Law

Snell's Law is a fundamental principle in optics that explains how light bends, or refracts, when it travels from one medium to another. This change of direction occurs due to the difference in the speed of light in different materials. Snell's Law is expressed by the formula:

• \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \)

Here, \( n_1 \) and \( n_2 \) are the indices of refraction of the two media, and \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction, respectively.
This law is crucial in understanding how light behaves when it hits a surface at an angle that is not perpendicular. In practical terms, it helps explain phenomena like why a swimming pool looks shallower than it actually is.
By using Snell's Law, we can find out how light bends when moving from air to water or vice versa, which is essential when determining apparent and actual depths.

apparent depth

Apparent depth is a concept that helps us understand how objects submerged in a transparent medium, like water, appear to be closer to the surface than they actually are. This optical illusion is due to the refraction of light rays as they move from water (a denser medium) to air (a less dense medium).
To calculate apparent depth, we use the formula:

• \( d_{apparent} = d_{actual} \times \frac{n_{water}}{n_{air}} \)

Where \( d_{apparent} \) is what appears to be the depth to an observer, \( d_{actual} \) is the true depth of the object, and \( n_{water} \) and \( n_{air} \) are the indices of refraction for water and air, respectively.
This formula is particularly useful in scenarios like swimming pools, where understanding the true positions of objects can be essential for safety and design purposes.

indices of refraction

Indices of refraction, or refractive indices, are numbers that describe how light propagates through a particular medium. The higher the index, the slower light travels within that medium.

• For air, the index of refraction is generally around \( 1.00 \), indicating that light travels at nearly its maximum speed.
• For water, the index is approximately \( 1.33 \), meaning light slows down by about 33% compared to in a vacuum.

These indices are crucial for calculating how much the light will bend when transitioning between mediums, as seen in Snell's Law. Knowing the indices of refraction helps in practical applications like designing lenses, understanding underwater imaging, and even in everyday scenarios like swimming pool design to correctly estimate object positions.