A) On a calm, bright day, you're scuba diving in a deep lake (n = 1.33), 6.00 m away from your sailboat in the horizontal direction. When you are 2.20 m below the surface, the top of the sailboat mast appears to you to be in a direction 36.0° from vertical. Calculate the height of the mast.
B)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.
A) To calculate the height of the mast, we can use the concept of refraction and trigonometry. Here are the steps:
Step 1: Draw a diagram with the sailboat, water surface, and the observer underwater. Let's label the observer as point O, the top of the mast as point T, and the point where the observer's line of sight intersects the water surface as point A.
Step 2: Consider the angles involved. The angle between the vertical line from the observer's eye and the observer's line of sight is 36.0°. Let's label this angle as θ.
Step 3: From Snell's Law, we know that the angle of refraction (r) can be related to the angle of incidence (i) and the refractive indices of the media involved (n1 and n2) using the formula: n1sin(i) = n2sin(r).
Step 4: Since the observer is looking from water into air, we have n1 = 1.33 (refractive index of water) and n2 = 1.00 (refractive index of air).
Step 5: Using the concept of refraction, the angle of incidence (i) can be calculated as the complement of the angle θ. So, i = 90° - θ.
Step 6: Now, we can use Snell's Law to find the angle of refraction (r). Rearranging the formula, we get: sin(r) = (n1 / n2) * sin(i).
Step 7: Substitute the values and calculate sin(r).
Step 8: Using trigonometry, we know that the height of the mast is given by the equation: height = distance * tan(r).
Step 9: Substitute the values of the distance and the calculated value of tan(r), then calculate the height of the mast.
B) To determine the diameter of the bright circle representing the sky, we can use the concepts of refraction and geometry. Here are the steps:
Step 1: Recall that the apparent depth of an object is given by the formula: apparent depth = real depth / refractive index.
Step 2: Since the observer is looking from water into air, the refractive index is 1.33.
Step 3: In this case, the object is the entire sky, and the real depth is the depth of the observer below the water surface (2.20 m).
Step 4: Using the formula, calculate the apparent depth of the observer below the water surface.
Step 5: The diameter of the bright circle representing the sky is twice the apparent depth of the observer. Multiply the calculated apparent depth by 2 to get the diameter of the circle.
To solve both questions, we can use the principles of geometric optics, specifically Snell's law and the concept of refraction.
A) To determine the height of the sailboat mast, we need to analyze the bending of light as it passes through the water-air interface. We can assume that the light travels in a straight-line path from the top of the mast to your eyes.
First, we need to find the angle of incidence of the light at the water-air interface. We know that the angle of incidence (measured from the normal to the interface) is the complement of the angle of elevation, which is 36.0°. Therefore, the angle of incidence is 90° - 36.0° = 54.0°.
According to Snell's law, the ratio of the trigonometric sines of the angles of incidence and refraction is equal to the ratio of the refractive indices of the two mediums:
sin(θ1) / sin(θ2) = n2 / n1
Where:
θ1 = angle of incidence
θ2 = angle of refraction
n1 = refractive index of medium 1 (water)
n2 = refractive index of medium 2 (air)
Given that n1 = 1.33 and n2 = 1 (since air has a refractive index close to 1), we can plug in the values:
sin(54.0°) / sin(θ2) = 1 / 1.33
Now we can solve for the angle of refraction:
sin(θ2) = sin(54.0°) * (1.33 / 1)
θ2 = arcsin(sin(54.0°) * (1.33 / 1))
The angle of refraction is approximately 67.0°.
Next, we can use basic trigonometry to find the height of the mast. The horizontal distance from you to the mast is given as 6.00 m, and the vertical distance you are below the surface is 2.20 m. Both of these distances form a right-angled triangle with the height of the mast.
tan(θ2) = height / distance
Therefore, height = tan(θ2) * distance
height = tan(67.0°) * 6.00 m
Using a calculator, we find that the height of the mast is approximately 15.32 m.
B) To determine the diameter of the bright circle that appears directly above you, we can use the concept of the critical angle.
The critical angle is the angle of incidence at which light passing from a denser medium (water) to a less dense medium (air) is refracted along the interface. For angles of incidence greater than the critical angle, total internal reflection occurs, causing the light to be reflected back into the water.
Since we are observing the light from below the water surface, the diameter of the bright circle will correspond to twice the distance from your point of observation to the point where the light rays are refracted at the critical angle.
To find the critical angle, we can use the following formula:
θc = arcsin(n2 / n1)
Given that n1 = 1.33 (water) and n2 = 1 (air), we find:
θc = arcsin(1 / 1.33)
Using a calculator, we find that the critical angle is approximately 48.8°.
Now, we can calculate the distance from your point of observation to the point where the light rays are refracted at the critical angle. This can be done using basic trigonometry:
distance = tan(θc) * depth
Given that you are 2.20 m below the surface, we have:
distance = tan(48.8°) * 2.20 m
Using a calculator, we find that the distance is approximately 2.48 m.
Finally, the diameter of the bright circle will be twice the distance, so the diameter is approximately 2 * 2.48 m = 4.96 m.