Q19E - University Physics with Modern Physics

Question

34.19 A person swimming 0.80cm below the surface of the water in a swimming pool looks at the diving board that is directly overhead and sees the image of the board that is formed by refraction at the surface of the water. This image is a height of 5.20 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 diving board above the surface of water is 3.30m.

Step 1: Formula for Spherical refracting surface with plane surface between two optical materials

$\frac{{\mathbf{n}}_{\mathbf{a}}}{\mathbf{s}}\mathbf{+}\frac{{\mathbf{n}}_{\mathbf{b}}}{\mathbf{s}\mathbf{\text{'}}}\mathbf{=}\mathbf{0}$

Step 2: Determine the actual height of diving board above the surface of water

There is a special case for the spherical refracting surface which is the plane surface between two materials where $R=\infty$. In this case, use the below equation for a plane refracting surface in the form

$\frac{n_a}{s}+\frac{n_b}{s\text{'}}=0$

Where $n_a$ is the refractive index of the air and equals 1, $n_b$ is the refractive index of water and equals 1.333. The distance s is the distance from the diving board to the person. $s\text{'}$ is the apparent depth and equals $s\text{'}=5.20m-0.80m=4.40m$. So, solve equation for s

$s=-\left(\frac{n_a}{n_b}\right)s\text{'}$                                                                                                                (1)

Put the values in above equation,

$s=-\left(\frac{n_a}{n_b}\right)s\text{'}=-\left(\frac{1.0}{1.333}\right)\left(-4.40m\right)=3.30m$

Hence, the actual height of diving board above the surface of water is 3.30m.