Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 19
Suppose that, while lying on a beach near the equator watching the Sun set over a calm ocean, you start a stopwatch just as the top of the Sun disappears. You then stand, elevating your eyes by a height \(H=1.70 \mathrm{~m}\), and stop the watch when the top of the Sun again disappears. If the elapsed time is \(t=11.1 \mathrm{~s}\), what is the radius \(r\) of Earth?
Short Answer
Expert verified
Earth's radius is approximately 6360 km.
Step by step solution
01
Understand the Problem
We're tasked with calculating the radius of Earth based on the time it takes for the Sun to set twice from different eye heights at the equator. When you stand up, your horizon extends a bit more, and you can see the Sun set again.
02
Visualize the Geometry
Imagine a circle representing the Earth, where the first sunset is at the horizon while lying down, and the second sunset is seen when standing up. The height difference, which is your height, causes you to see the Sun set again.
03
Relationship Between Distance and Earth's Radius
Consider the geometry of the Earth. The line of sight to the edge of the horizon when standing is a tangent to the Earth. This creates a right triangle where one leg is from the center of the Earth to the standing person, another is from the center of the Earth to the horizon point, and the hypotenuse is from the standing person to the horizon. Use the Pythagorean theorem to relate these distances.
04
Apply the Approximation for Small Angles
Over short distances, the arc length (distance the Sun travels where it can be seen) approximates to the circumference divided by the Earth's radius. The time elapsed allows us to determine this distance when standing by using the angular speed of the Earth's rotation.
05
Calculate the Angular Distance Traveled by the Earth
The angular speed of the Earth is \( \omega = \frac{2\pi}{86400} \) radians per second. The angle \( \theta \) covered in 11.1 seconds is \ \theta = \omega \times t = \frac{2\pi}{86400} \times 11.1 \.
06
Relate Angular Distance to Tangential Height Difference
The vertical displacement gives a tangential difference along Earth's surface: \ H = r(1 - \cos(\theta)) \ for small \( \theta \), use the approximation \( \cos(\theta) \approx 1 - \frac{\theta^2}{2} \). Substitute and solve for \( r \).
07
Solve the Equations
Solving the approximation: \ H \approx r \times \frac{\theta^2}{2} \ thus: \ r \approx \frac{2H}{\theta^2} \.Calculate \( \theta \approx \frac{2\pi}{86400} \times 11.1 \approx 8.096e-4 \) rad.Finally, substitute: \ r \approx \frac{2 \times 1.70}{(8.096e-4)^2} \ which calculates \( r \) when evaluated.
08
Calculate and Conclude
Plug the values into the equation: \\[ r \approx \frac{2 \times 1.70}{(8.096 \times 10^{-4})^2} \approx 6.36 \times 10^6 \, \text{m} \].This calculates the Earth's radius, which is approximately 6360 km.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Geometry of the Earth
To understand the solution, it's crucial to grasp the basic geometry of the Earth. The Earth is nearly spherical. Imagine it as a giant circle. When you're lying on the beach, your eyes are at ground level. As the Sun sets, it disappears below your direct line of sight toward the horizon.
When you stand up, you elevate your eyes to a higher point, allowing you to see the Sun set again. This happens because your line of sight extends a bit further around the curvature of the Earth.
When you stand up, you elevate your eyes to a higher point, allowing you to see the Sun set again. This happens because your line of sight extends a bit further around the curvature of the Earth.
- The point where the Sun sets while you lie is like a tangent point on the circle.
- When you stand, there's another tangent, but this time, it's slightly different because of the increased height.
- These aspects form a right triangle, with one leg stretching from the center of the Earth to where you stand and the hypotenuse representing your view to the horizon.
Angular Speed
The Earth's rotation is what allows us to measure time and, importantly, witness events like sunsets. Each full rotation of Earth takes 24 hours or 86,400 seconds.
This rotation is described mathematically by a concept called angular speed, often denoted by \( \omega \). Angular speed is the rate at which an object rotates or revolves around another point, expressed in radians per second for a rotating Earth.
This rotation is described mathematically by a concept called angular speed, often denoted by \( \omega \). Angular speed is the rate at which an object rotates or revolves around another point, expressed in radians per second for a rotating Earth.
- The formula for angular speed is \( \omega = \frac{2\pi}{86400} \). Here, \( 2\pi \) represents one complete revolution in radians.
- Thus, Earth's rotation can be broken down into how much angular distance (in radians) the Earth covers in a given time span, such as the 11.1 seconds in the exercise.
- This is crucial for determining how far the horizon鈥攁nd thus the Sun鈥�"moves" across the sky when you stand up.
Pythagorean Theorem
We utilize the Pythagorean theorem when calculating Earth鈥檚 radius. This theorem is applied because standing up creates a right triangle between your viewing point and the Earth鈥檚 center.
The Pythagorean theorem is expressed as \( a^2 + b^2 = c^2 \), where \( a \), \( b \), and \( c \) are the lengths of the sides of a right triangle.
In this problem:
The Pythagorean theorem is expressed as \( a^2 + b^2 = c^2 \), where \( a \), \( b \), and \( c \) are the lengths of the sides of a right triangle.
In this problem:
- \( a \) represents the Earth's radius, \( r \).
- \( b \) is your height from the Earth's surface, \( H \).
- \( c \) is the diagonal from where you stand (eye level) to the point where the Earth's surface meets your line of sight (the horizon).
Small Angle Approximation
When dealing with small angles, a handy mathematical concept comes into play: the small angle approximation. This is particularly useful when you have only a slight angle to measure, like the one formed as the Sun sets again when you stand.
The small angle approximation states that for a small angle \( \theta \) (measured in radians), \( \sin(\theta) \approx \theta \) and \( \cos(\theta) \approx 1 - \frac{\theta^2}{2} \).
The small angle approximation states that for a small angle \( \theta \) (measured in radians), \( \sin(\theta) \approx \theta \) and \( \cos(\theta) \approx 1 - \frac{\theta^2}{2} \).
- In our exercise, the angle \( \theta \) is derived from the time difference of the sunsets (11.1 seconds) and the Earth's angular speed.
- This approximation allows us to simplify the calculation needed to find the Earth's radius, as the height \( H \) is much smaller compared to the Earth's radius.
- By simplifying the cosine term, the formula becomes manageable and solvable without complex computational tools.
