On a calm, bright day, you're scuba diving in a deep lake (n = 1.33), 5.40 m from your sailboat. When you are 3.10 m below the surface, the top of the sailboat mast appears to you to be in a direction 40.0° from vertical. Calculate the height of the mast
part two
From your point of view, the entire sky appears to be confined to a bright disk directly above you on the surface of the water. Determine the diameter of this circle.
I draw a picture and i can use snell law but i am stuck at the trig part. Or my drawing is wrong help
above answer is not correct
Let's break down the problem step-by-step:
Step 1: Calculate the angle of incidence
Given that the mast appears to be 40.0° from vertical, we can calculate the angle of incidence using Snell's law. The formula for Snell's law is:
n₁ sinθ₁ = n₂ sinθ₂
Here, n₁ is the refractive index of air (approximated as 1.00) and n₂ is the refractive index of water (given as 1.33). θ₁ is the angle of incidence (unknown), and θ₂ is the angle between the vertical direction and the light ray underwater (40.0°).
Plugging in the values, we have:
1.00 sin(θ₁) = 1.33 sin(40.0°)
To find θ₁, we can rearrange the equation:
sin(θ₁) = (1.33/1.00) sin(40.0°)
θ₁ = sin^(-1)(1.33/1.00 * sin(40.0°))
θ₁ ≈ 56.6°
Step 2: Calculate the distance from the surface to the mast
To find the height of the mast, we need to determine the vertical distance between your position and the top of the mast. This distance can be split into two components:
- The depth you are below the surface (3.10 m)
- The height from the surface to the top of the mast (unknown)
Using trigonometry, we can relate the two components:
tan(θ₁) = height of mast / depth below the surface
Plugging in the values:
tan(56.6°) = height of mast / 3.10 m
Solving for the height of the mast:
height of mast = 3.10 m * tan(56.6°)
height of mast ≈ 4.41 m
So the height of the mast is approximately 4.41 meters.
Now let's move on to the second part:
Step 1: Calculate the angle between the water surface and the top of the sky disk
Since the sky appears confined to a bright disk directly above the water's surface, the angle between the vertical direction and the top edge of the disk is the same as the angle of incidence (θ₁ from above), which we found to be 56.6°.
Step 2: Calculate the diameter of the circle on the water's surface
To determine the diameter of the circle, we need to find the distance between you (underwater) and the edge of the disk.
Again, we can use trigonometry:
tan(θ₁) = radius of the circle / distance from your position to the edge of the circle
Given that you are 5.40 m away from the sailboat, we can plug in the values:
tan(56.6°) = radius of the circle / 5.40 m
Solving for the radius of the circle:
radius of the circle = 5.40 m * tan(56.6°)
radius of the circle ≈ 7.80 m
Finally, the diameter of the circle would be twice the radius:
diameter of the circle ≈ 2 * 7.80 m
diameter of the circle ≈ 15.6 m
So, the diameter of the circle on the water's surface is approximately 15.6 meters.
To calculate the height of the mast, we can use the principles of trigonometry and Snell's Law.
Let's start by visualizing the situation. You are underwater, and the sailboat mast appears at an angle of 40.0° from the vertical. The distance between you and the sailboat is given as 5.40 m. We can represent this situation with a right triangle, where the height of the mast is the opposite side, the distance between you and the mast is the adjacent side, and the angle is the angle of elevation.
To calculate the height of the mast, we can use the tangent function:
tan(angle) = opposite / adjacent
In this case, the angle is 40.0°, and the adjacent side is 5.40 m. Let's solve for the opposite side, which represents the height of the mast:
height = tan(angle) * distance
height = tan(40.0°) * 5.40 m
Using a scientific calculator, we can calculate:
height ≈ 3.82 m
Therefore, the height of the mast is approximately 3.82 meters.
Now, let's move on to the second part of your question. You mention that the entire sky appears confined to a bright disk directly above you on the surface of the water. This situation can be visualized as a circle with the observer (you) at the center and the surface of the water as the circumference of the circle.
To determine the diameter of this circle, we need to take into account the refractive index of water, which is given as n = 1.33. The refraction of light at the air-water interface causes a change in the apparent size and position of objects.
To calculate the diameter of the circle, we need to consider the apparent angle of the circle from your point of view, which is the angle between the vectors representing the vertically above and the horizon on the water's surface.
We can use Snell's Law to relate the angles of incidence (from vertical) and refraction:
n1 * sin(angle of incidence) = n2 * sin(angle of refraction)
In this case, n1 is the refractive index of air (approximately 1) and n2 is the refractive index of water (1.33). The angle of incidence is 90°, as the circle appears vertically above you, and we want to find the angle of refraction.
sin(angle of refraction) = (n1 / n2) * sin(angle of incidence)
sin(angle of refraction) = (1 / 1.33) * sin(90°)
Using a scientific calculator, we can calculate the sine of 90° as 1:
sin(angle of refraction) = (1 / 1.33) * 1
sin(angle of refraction) ≈ 0.75
Now, we can use the inverse sine (or arcsine) function on our calculator to find the angle of refraction:
angle of refraction ≈ arcsine(0.75)
angle of refraction ≈ 48.6°
The angle of refraction gives us the angle from the vertical to the apparent top edge of the circle.
Now, we can use the tangent function to find the angle from the vertical to the apparent bottom edge of the circle:
tan(angle from vertical to bottom edge) = tan(angle of refraction) / 2
Using a scientific calculator, we can calculate:
tan(angle from vertical to bottom edge) ≈ tan(48.6°) / 2
tan(angle from vertical to bottom edge) ≈ 0.85 / 2
tan(angle from vertical to bottom edge) ≈ 0.425
Now, we can calculate the angle between the apparent top and bottom edges of the circle:
angle between top and bottom = 90° - 2 * angle from vertical to bottom edge
Using a scientific calculator, we can calculate:
angle between top and bottom ≈ 90° - 2 * arcsin(0.425)
angle between top and bottom ≈ 59.7°
Finally, we can use the tangent function to calculate the diameter of the circle:
diameter = 2 * height * tan(angle between top and bottom)
Using the height we previously calculated (approximately 3.82 m), we can solve for the diameter:
diameter = 2 * 3.82 m * tan(59.7°)
Using a scientific calculator, we can calculate:
diameter ≈ 2 * 3.82 m * 1.687
diameter ≈ 12.88 m
Therefore, the diameter of the circle that appears as the entire sky from your point of view on the water's surface is approximately 12.88 meters.
Do you need help on part one or part two?
Part 1: The ray from the top of the mast that you see has an angle of incidence of
tan^-1 = H/5.40. Tha angle of refraction is tan ^-1 3.10/5.40 = 29.9 degrees
Now use Snell's law:
sin [tan^-1(H/5.40)] = 1.33 sin 29.9
Solve for H
sin^[tan^-1(H/5.40)] = 0.662
tan^-1(H/5.40) = 41.4 degrees
H/5.40 = 0.884
H = 4.77 m
For part 2, assume that the angle of incidence for the sky above is 90 degrees at the edge of the circular disc. Solve for the angle of refraction and use that to get the diameter of the circle of light