Question

After a long day of driving you take a late-night swim in a motel swimming pool. When you go to your room, you realize that you have lost your room key in the pool. You borrow a powerful flashlight and walk around the pool, shining the light into it. The light shines on the key, which is lying on the bottom of the pool, when the flashlight is held y = 1.2 m above the water surface and is directed at the surface a horizontal distance of x = 1.8 m from the edge (see figure below). If the water here is d = 5.0 m deep, how far is the key from the edge of the pool?

Answer 1

Concepts and reason

The concepts used to solve this problem are Snell’s law and trigonometric equations

Firstly, use the trigonometric equation relation between the distance at which the flashlight is held and the horizontal distance to calculate the angle of incidence.

Use the Snell’s law relation between the angle of incidence and refractive index of the water and sir to calculate the angle of refraction.

First use the trigonometric equation relation between the angle of incidence and the depth of the water to calculate the distance from the horizontal distance to the spot of the flashlight.

Finally, add the horizontal distance and distance from the horizontal distance to the spot of the flashlight to distance of the key from the edge of the pool.

Fundamentals

Free body diagram of the given figure,

From the free body diagram, the expression for the is,

Here, is the angle of incidence, is the distance at which the flashlight is held, and is the horizontal distance.

Expression for the Snell’s law is,

Here, is the refractive index of air, is the refractive index of water, and is the angle of refraction.

Expression for the is,

Here, is the distance from the horizontal distance to the spot of the flashlight and is the depth of the water.

Expression for the total distance of the key from the edge of the pool is,

Here, d is the total distance of the key from the edge of the pool.

Expression for the is,

Substitute for y and for x.

Rearrange the above equation to get the angle of incidence,

Expression for the Snell’s law is,

Substitute for , for , and for .

Rearrange the above equation to the angle of refraction,

Expression for the is,

Substitute for and for h.

Rearrange the above equation to get the distance from the horizontal distance to the spot of the flashlight,

Expression for the total distance of the key from the edge of the pool is,

Substitute for s and for x.

Ans:

The distance of the key from the edge of the pool is .