You are lying on the beach on a tropical island on the equator, watching the Sun set - what a beautiful scene! Immediately after the top of the Sun disappears, you stand up to watch the Sun set again. After the amount of time t=10 s the top of the Sun disappears again. Given that you are about h=1.8 m tall, estimate the radius of the Earth in m.

Question

You are lying on the beach on a tropical island on the equator, watching the Sun set - what a beautiful scene! Immediately after the top of the Sun disappears, you stand up to watch the Sun set again. After the amount of time t=10 s the top of the Sun disappears again. Given that you are about h=1.8 m tall, estimate the radius of the Earth in m.

Answer 1

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Answer 2

To estimate the radius of the Earth using the given information, we can use the concept of angular velocity.

When the Sun sets, it appears to move in an arc across the sky due to the rotation of the Earth. The time it takes for the Sun to move from its highest point in the sky to the horizon is approximately 12 hours or 43,200 seconds. This corresponds to a full 360-degree rotation.

Let's consider the first sunset you observed. As the top of the Sun disappears, you stand up to watch the sunset again. This means you raise your viewpoint by your height, which is h = 1.8 m. In a time interval of t = 10 s, you observe the Sun disappear for the second time.

From this information, we can calculate the angular velocity of the Earth's rotation using the equation:

Angular Velocity (ω) = Change in Angle / Time Interval

Since the Sun has moved by an angle equivalent to your change in height, we can write:

Angular Velocity (ω) = (2π) / t

where t is the time interval in seconds.

Now, using the angular velocity, we can find the circumference of the Earth using the equation:

Circumference = Angular Velocity * Radius

Since a full rotation is 360 degrees or 2π radians, the Earth's circumference is equal to 2π times its radius.

Using these equations, we can rearrange the formulas to solve for the radius of the Earth:

Radius = Circumference / (2π)

Substituting angular velocity into the circumference equation:

Radius = (Angular Velocity * Radius) / (2π)

Simplifying the equation:

Radius = (Angular Velocity * Radius) / (2π)

Radius = (2π * h) / (2π * t)

Canceling out the πs, we get:

Radius = (h / t)

Substituting the given values:

Radius = (1.8 m / 10 s)

Calculating:

Radius = 0.18 m

Therefore, the estimated radius of the Earth is approximately 0.18 meters.