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?
See my later answer to the same question.
To find the length of the shadow on the floor of the pool, we can use the principles of similar triangles and Snell's Law.
First, let's consider the situation outside of the pool. The stick is 1.20 meters long, and it casts a shadow of the same length. This means that the ratio of the length of the stick to the length of its shadow is 1:1.
Now, let's examine the situation inside the pool. The stick is now partly submerged in saltwater, which has a refractive index (n) of 1.71. When light travels from air to water, it bends or refracts. This means that the light rays coming from the stick will change direction as they pass from air into water.
To calculate how the light rays refract, we can use Snell's Law, which states:
n1 * sin(theta1) = n2 * sin(theta2),
Where n1 is the refractive index of the initial medium (in this case, air), theta1 is the angle of incidence, n2 is the refractive index of the final medium (in this case, saltwater), and theta2 is the angle of refraction.
In this scenario, the stick is held vertically against the flat bottom of the pool, so the light rays are incident perpendicular to the bottom of the pool (i.e., theta1 = 0 degrees). Therefore, the equation simplifies to:
n1 * sin(0) = n2 * sin(theta2),
0 = n2 * sin(theta2).
Since sin(0) = 0, the angle of refraction (theta2) is also 0, meaning the light rays continue along the same path in the water, without bending.
So, when the light rays hit the floor of the pool, they will form the same angle with the floor as they did outside of the pool. Therefore, the length of the shadow on the floor of the pool will be the same as the length of the stick, which is 1.20 meters.