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 2. 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.
Part 1. Draw the figure, apply Snell's law. Use trig to gett the angles.
To calculate the height of the mast, we can use the principles of optics and trigonometry. Here's how you can approach it:
Step 1: Determine the angle of incident and the angle of refraction:
The angle of incident is the angle between the incident ray (a straight line drawn from the top of the mast to your eye) and the normal (a line perpendicular to the surface of the water at the point of entry). In this case, the angle of incident is 40.0° from the vertical.
The angle of refraction is the angle between the refracted ray (a straight line drawn from the point on the mast where the light enters the water and to your eye) and the normal. This angle can be calculated using Snell's law, which states: n1*sinθ1 = n2*sinθ2, where n1 and n2 are the refractive indices of the two mediums, and θ1 and θ2 are the angles of incidence and refraction, respectively.
In this case, the refractive indices are n1 = 1 (air) and n2 = 1.33 (water). Since the angle of incidence is known to be 40.0°, we can plug these values into Snell's law and solve for the angle of refraction (θ2).
Step 2: Find the distance from your eye to the point where the light enters the water:
To find this distance, we need to use the concept of similar triangles. The height of the mast can be considered as the opposite side, and the distance between your eye and the point where the light enters the water can be considered as the adjacent side of the triangle formed. So, we can use the tangent function to find this distance.
Let's call this distance D.
Step 3: Calculate the height of the mast:
Now that we have the distance D, we can use it to find the height of the mast using the tangent function again. The height of the mast can be considered as the opposite side, and the distance D can be considered as the adjacent side of the triangle formed.
Let's call the height of the mast H.
Using the tangent function, we have: tan(θ2) = H/D
Rearranging the equation, we can solve for H:
H = D * tan(θ2)
Once you calculate the value of H, you will have the height of the mast.
Now, to determine the diameter of the circle representing the entire sky from your point of view, we can make some observations.
Since the surface of the water acts as a mirror due to total internal reflection, the reflection of the sky will appear as a circle on the water's surface. The diameter of this circle will correspond to the maximum possible viewing angle.
In this case, the diameter can be calculated by using the concept of the critical angle and the refractive index of water. The critical angle is the angle of incidence that results in an angle of refraction of 90°.
Step 1: Find the critical angle:
Using the formula for the critical angle, which is sin(critical angle) = 1/n, where n is the refractive index of water, we can solve for the critical angle.
In this case, the refractive index of water is given as 1.33. Substitute the value of n into the formula to find the critical angle.
Step 2: Calculate the diameter of the circle:
The diameter of the circle representing the entire sky is twice the product of the distance between your eye and the point where the light enters the water and the tangent of the critical angle.
Let's call this diameter D_sky.
D_sky = 2 * D * tan(critical angle)
Once you calculate the value of D_sky, you will have the diameter of the circle representing the entire sky.