If an observer at a seashore measures the time lapse between two sunsets by observing the Sun from two different heights above sea level, which of the following physical quantities about Earth or the environment can be estimated from this time difference?

Introduction / Context:
This question connects a simple observation at a beach with a deeper concept in astronomy and geometry. It tests whether you know that by measuring the time difference between sunsets observed from two different heights, one can estimate the radius of the Earth using basic rotational motion and geometry.

Given Data / Assumptions:
• An observer watches sunset from one height near sea level and then from a higher point, such as a tall building or hill near the beach.
• There is a measurable time lapse between the two sunsets because the higher observer sees the Sun for slightly longer.
• The Earth rotates at a nearly constant angular speed, and the Sun is effectively at a very large distance compared to Earth radius.

Concept / Approach:
When the observer is at a greater height, the line of sight to the Sun reaches over the Earth curvature a bit longer. The time difference between the two sunsets corresponds to a small additional angle of rotation of the Earth. By relating this angle to the height of the observer and the geometry of a circle, you can derive the radius of the Earth. Other quantities like the distance to the Sun or the radius of the Sun require more complex observations and are not determined by such a simple beach experiment.

Step-by-Step Solution:
Step 1: Recognise that the Earth rotates once in about 24 hours, which is 360 degrees of rotation. Step 2: If the time difference between the two sunsets is measured as a few seconds or minutes, this corresponds to a small additional angle of rotation. Step 3: At a higher height, the horizon dips slightly, so the observer can see the Sun for a bit longer before it disappears behind the curved Earth. Step 4: Using simple right triangle geometry with the Earth radius and observer height, this angle and the height are related through the geometry of a circle. Step 5: By combining the height and the angle derived from the time lapse, one can calculate an approximate value of the Earth radius.

Verification / Alternative check:
Historically, methods based on horizon distance and timing have been used to approximate the Earth radius. For example, measuring how far you can see to the horizon from different heights yields similar geometric relationships. None of these simple observations can directly give the distance to the Sun or the radius of the Sun, which require more advanced astronomical techniques, so radius of Earth is the only realistic outcome here.

Why Other Options Are Wrong:
Distance between the Sun and the Earth: This requires more complex astronomical measurements and cannot be deduced from a local difference in sunset times at two heights.
Depth of the ocean at that beach: Ocean depth does not significantly affect sunset time, which is mainly determined by Earth curvature and rotation.
Radius of the Sun: The apparent size of the Sun in the sky is not directly linked to this short time difference at the horizon.
Height of the atmosphere above the Earth: While refraction can slightly affect apparent sunrise and sunset times, precise atmospheric height estimation demands more detailed data than a simple two height observation provides.

Common Pitfalls:
Students often think of the dramatic scale of the Sun and may wrongly assume that sunrise and sunset observations directly give the Sun distance. Another mistake is to overlook the role of Earth curvature. Remember that the key reason a higher observer sees the Sun for longer is that the Earth is round, and exploiting that geometry lets us estimate the radius of the Earth, not the properties of the Sun itself.

Final Answer:
From the time lapse between two sunsets viewed at different heights above sea level, one can estimate the radius of the Earth.