At 08:00hrs a 2.0- m-long vertical stick in air casts a shadow 1.8 m long. If the same stick is placed at 08:00hrs in air in a flat bottomed pool of salt water half the height of the stick, how long is the shadow on the floor of the pool? (For this pool, n = 1.56.)
To find the length of the shadow on the floor of the pool, we need to use the concept of similar triangles. Let's break down the problem step by step:
1. We are given that at 08:00hrs, a 2.0-m-long stick in air casts a shadow 1.8 m long. This means the ratio of the height of the stick to the length of its shadow is 2.0/1.8, or 10/9.
2. Now, if we place the same stick at 08:00hrs in a flat-bottomed pool of salt water, we need to consider the refractive index of the water. In this case, the refractive index (n) is given as 1.56.
3. According to the laws of refraction, when light passes from air to water, it bends and changes direction. This bending of light causes the apparent position of the stick to be higher, giving an impression that the stick is shorter.
4. To account for the bending of light, we need to consider the effective height of the stick in water. The effective height (he) can be calculated using the formula he = h/n, where h is the actual height of the stick and n is the refractive index of the medium (in this case, the water).
5. Plugging in the values, he = 2.0 m / 1.56 = 1.28 m.
6. Now that we have the effective height of the stick in the water, we can determine the length of the shadow on the pool floor. Since the stick is half-submerged, the length of the shadow will be proportional to the effective height. We previously found that the ratio of the height of the stick to the length of its shadow in air was 10/9. Therefore, we can set up a proportion:
(1.28 m - 1.0 m) / x = 10/9
In this proportion, x represents the length of the shadow on the pool floor.
7. Solving the proportion:
1.28 m - 1.0 m = (10/9) * x
0.28 m = (10/9) * x
x = (0.28 m) * (9/10) ≈ 0.252 m
Therefore, the length of the shadow on the floor of the pool is approximately 0.252 meters.