An underwater diver sees the sun 50° above horizontal. How high is the sun above the horizon to a fisherman in a boat above the diver?

검증된 단계별 안내

Step 1: Understand the concept of refraction. When light passes from one medium to another (e.g., water to air), its speed changes, causing the light to bend. This bending is described by Snell's Law:

n₁ sin θ₁ = n₂ sin θ₂

, where n₁ and n₂ are the refractive indices of the two media, and θ₁ and θ₂ are the angles of incidence and refraction, respectively.

Step 2: Identify the refractive indices for water and air. The refractive index of water is approximately

n₁ = 1.33

, and the refractive index of air is approximately

n₂ = 1.00

. These values will be used in Snell's Law.

Step 3: Recognize that the angle given (50°) is the angle of refraction,

θ₂

, as seen by the diver underwater. You need to calculate the angle of incidence,

θ₁

, which is the angle of the sun above the horizon as seen by the fisherman in the boat.

Step 4: Rearrange Snell's Law to solve for

θ₁

θ₁ = arcsin((n₂ / n₁) * sin θ₂)

. Substitute the known values:

n₁ = 1.33

n₂ = 1.00

, and

θ₂ = 50°

Step 5: Perform the calculation using the formula derived in Step 4. This will give you the angle of incidence,

θ₁

, which represents the sun's height above the horizon as seen by the fisherman.

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주요 개념

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Angle of Elevation

The angle of elevation is the angle formed by the horizontal line and the line of sight to an object above that line. In this scenario, the diver sees the sun at an angle of 50° above the horizontal, which indicates how high the sun appears from the diver's perspective underwater.

Refraction of Light

Refraction is the bending of light as it passes from one medium to another, such as from water to air. This phenomenon affects how objects appear to observers in different mediums, meaning the angle at which the fisherman sees the sun may differ from the angle perceived by the diver due to the change in medium.

Geometry of Sight Lines

The geometry of sight lines involves understanding how angles and distances relate to the positions of observers and objects. In this case, both the diver and the fisherman are at different depths, and the geometry helps determine the actual height of the sun above the horizon as seen by the fisherman based on the diver's angle of elevation.