On a sunny day, a 1.20 m long vertical stick in air casts a shadow 1.20 m long. If the same stick is held vertically and touching the flat bottom of a pool of salt water (n=1.71) half the height of the stick, how long is the shadow on the floor of the pool?
The angle of incidence above the water is 45 degrees, because of the equal stick and shadow lengths above the water.
Then with half the stick above the water, the top of the stick's shadow hits the water 0.60 m from the stick. Then it proceeds below the water at a smaller angle of refraction given by Snell's law.
0.60 m times the tangent of the refraction angle (added to 0.60 m) will tell you where it hits the bottom of the pool.
The index of refraction (1.71) that they tell you to use for saltwater is much too high, but they probably expect you to use it anyway.
To determine the length of the shadow on the floor of the saltwater pool, we can use the principles of geometric optics and Snell's Law.
First, let's consider the situation on a sunny day when the stick is in the air. In this case, the stick's height and the length of its shadow are both 1.20 m.
Now, when the stick is submerged in the saltwater pool, we need to take into account the refractive index of saltwater, which is given as n = 1.71.
Using Snell's Law, we can relate the angles of incidence and refraction at the water-air interface:
n1 * sin(θ1) = n2 * sin(θ2)
Here:
n1 = refractive index of the medium the light is coming from (air) = 1
θ1 = angle of incidence of the light ray
n2 = refractive index of the medium the light is entering (saltwater) = 1.71
θ2 = angle of refraction of the light ray
Since the stick is held vertically, the incident light ray (coming from the top) will be perpendicular to the water-air interface, which means θ1 = 90 degrees.
We can now calculate the angle of refraction θ2 using Snell's Law:
1.71 * sin(θ2) = 1 * sin(90)
sin(θ2) = sin(90) / 1.71
θ2 = arcsin(1 / 1.71)
Now, let's move on to finding the length of the shadow on the floor of the pool. When the light exits the water-air interface and reaches the pool floor, the angle of incidence will be equal to the angle of refraction, which is θ2.
Since the stick is half the height (0.60 m) underwater, we need to consider the geometry of the situation. The length of the shadow on the floor will be equal to the length of the portion of the stick submerged in the water.
Given that the total length of the stick is 1.20 m, and half of it is underwater, the length of the shadow on the floor will be:
Length of shadow = length of stick * (length of stick underwater / total length of stick)
Length of shadow = 1.20 m * (0.60 m / 1.20 m)
Simplifying, we find:
Length of shadow = 0.60 m
Therefore, the shadow on the floor of the saltwater pool will be 0.60 meters long.