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Chapter 1: Problem 45

The circumference of a circle is given by \(C=2 \pi r,\) where \(r\) is the radius of the circle. a. Calculate the approximate circumference of Earth's orbit around the Sun, assuming that the orbit is a circle with a radius of \(1.5 \times 10^{8} \mathrm{km}\). b. Noting that there are 8,766 hours in a year, how fast, in kilometers per hour, does Earth move in its orbit? c. How far along in its orbit does Earth move in 1 day?

Short Answer

Expert verified
Circumference: \(9.42 \times 10^{8} \text{km}\). Speed: \(1.074 \times 10^{5} \text{km/h}\). Distance in 1 day: \(2.578 \times 10^{6} \text{km}\).

Step by step solution

01

Identify the formula for circumference

The formula to calculate the circumference of a circle is given by \[ C = 2 \pi r \] where \( r \) is the radius.
02

Plug in the radius of Earth's orbit

Given the radius of Earth's orbit is \( 1.5 \times 10^{8} \text{km} \), substitute \( r \) in the formula:\[ C = 2 \pi (1.5 \times 10^{8}) \]
03

Calculate the approximate circumference

Use \( \pi \approx 3.14 \) to simplify the calculation:\[ C \approx 2 \times 3.14 \times 1.5 \times 10^{8} \]\[ C \approx 9.42 \times 10^{8} \text{km} \]
04

Find the speed of Earth's orbit in kilometers per hour

Since there are 8,766 hours in a year, divide the circumference by the number of hours to get the speed:\[ \text{Speed} = \frac{C}{8,766} \approx \frac{9.42 \times 10^{8}}{8,766} \]\[ \text{Speed} \approx 1.074 \times 10^{5} \text{km/h} \]
05

Calculate the distance Earth travels in 1 day

There are 24 hours in a day. Multiply the speed by the number of hours in a day:\[ \text{Distance in 1 day} = \text{Speed} \times 24 \approx 1.074 \times 10^{5} \times 24 \]\[ \text{Distance in 1 day} \approx 2.578 \times 10^{6} \text{km} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

circumference of a circle
To understand the Earth's orbit, it's essential to grasp the concept of the circumference of a circle. The circumference is the total distance around a circle. We use the formula \( C = 2 \pi r \) where \( r \) is the radius.

For example, imagine a circle with a radius of 2 km. The circumference would be \( 2 \pi \times 2 \), which simplifies to around 12.56 km using \( \pi \approx 3.14 \).

This formula allows us to calculate the circumference of any circle, regardless of size, simply by knowing its radius.
Earth's orbit radius
The radius of Earth's orbit is the distance from the Earth to the Sun. This distance is approximately \( 1.5 \times 10^8 \) kilometers.

Just like with any circle, knowing the radius lets us calculate the circumference of Earth's orbit.

When applying the formula \( C = 2 \pi r \), we plug in \( 1.5 \times 10^8 \) for \( r \), making the calculation: \( C \approx 2 \times 3.14 \times 1.5 \times 10^8 = 9.42 \times 10^8 \) km.

This means Earth's orbit has an immense circumference, about 942 million kilometers!
speed of Earth's orbit
Knowing the circumference of Earth's orbit helps us find out how fast our planet travels around the Sun. To find the speed, we need to know the total distance traveled in one year and how many hours are in a year.

There are 8,766 hours in a year. So, if we take the circumference from the previous calculation, we can determine the speed as follows: \( \text{Speed} = \frac{9.42 \times 10^8}{8,766} = 1.074 \times 10^5 \) km/h.

This means Earth travels roughly 107,400 kilometers every hour on its journey around the Sun.
distance traveled in a day
Now that we know Earth's speed in its orbit, we can calculate how far it travels in one day. There are 24 hours in a day.

By multiplying the speed by the number of hours in a day, we get the distance: \( \text{Distance in 1 day} = 1.074 \times 10^5 \times 24 = 2.578 \times 10^6 \) km.

This calculation shows that Earth travels about 2,578,000 kilometers every day, giving us a better understanding of the incredible distances involved in our planets orbit.

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Most popular questions from this chapter

A textbook published in 1945 stated that light takes 800,000 years to reach Earth from the Andromeda Galaxy. In this book, we assert that it takes 2,500,000 years. What does this difference tell you about a scientific "fact" and how our knowledge evolves with time?

Throughout this book, we will examine how discoveries in astronomy and space are covered in the media. Go to your favorite news website (or to one assigned by your instructor) and find a recent article about astronomy or space. Does this website have a separate section for science? Is the article you selected based on a press release, on interviews with scientists, or on an article in a scientific journal? Use Google News or the equivalent to see how widespread the coverage of this story is. Have many newspapers carried it? Has it been picked up internationally? Has it been discussed in blogs? Do you think this story was interesting enough to be covered?

The distance from Earth to Mars varies from 56 million km to 400 million \(\mathrm{km} .\) How long does it take a radio signal traveling at the speed of light to reach a spacecraft on Mars when Mars is closest and when Mars is farthest away?

Imagine that you have become a biologist, studying rats in Indonesia. Most of the time, Indonesian rats maintain a constant population. Every half century, however, these rats suddenly begin to multiply exponentially! Then the population crashes back to the constant level. Sketch a graph that shows the rat population over two of these episodes.

New York is 2,444 miles from Los Angeles. What is that distance in car-hours? In car-days? (Assume a travel speed of \(70 \mathrm{mph} .)\)
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