Problem 73 - Algebra and Trigonometry

Question

Orbit of the Earth Find the distance that the earth travels in one day in its path around the sun. Assume that a year has 365 days and that the path of the earth around the sun is a circle of radius 93 million miles. IThe path of the earth around the sun is actually an ellipse with the sun at one focus (see Section \(11.2 ) .\) This ellipse, however, has very small eccentricity, so it is nearly circular.]

Step-by-Step Solution

Answer

Distance traveled by Earth in one day is about 1,600,000 miles.

Step 1: Define the Problem

We need to find the distance the Earth travels in one day around the Sun, assuming the Earth's orbit is a circle. Given that a year is 365 days and the radius of the Earth's orbit is 93 million miles, we will calculate the circumference of this circle to find the total distance traveled in a year.

Step 2: Calculate the Circumference of the Earth's Orbit

The circumference of a circle can be determined by the formula \[ C = 2 \pi r \]where \( r \) is the radius of the circle. Substituting \( r = 93 \times 10^6 \) miles, we have:\[ C = 2 \pi \times 93 \times 10^6 \approx 2 \times 3.14 \times 93 \times 10^6 \approx 584 \times 10^6 \text{ miles} \]

Key Concepts

Circumference Calculation

Understanding how to calculate the circumference is a critical skill when dealing with circular paths. The circumference is the total distance around a circle and can be found using the formula:

• \( C = 2 \pi r \)

where \( C \) is the circumference and \( r \) is the radius of the circle.
In our exercise, the radius \( r \) is given as 93 million miles.
By substituting \( r = 93 \times 10^6 \) miles into the formula, we calculate as follows:

• \( C = 2 \pi \times 93 \times 10^6 \)
• Approximating \( \pi \) as 3.14, this becomes \( C \approx 2 \times 3.14 \times 93 \times 10^6 \)
• This simplifies to \( C \approx 584 \times 10^6 \text{ miles} \)

Therefore, the circumference of Earth's orbit is approximately 584 million miles, which is the total distance Earth travels in a full orbit around the Sun.

Circular Motion

Circular motion refers to the motion of an object along the circumference of a circle or a circular path.
In our scenario involving Earth's orbit, we assume a perfect circular path for simplification.
This model helps in easily understanding how Earth rotates around the Sun. Circular motion involves velocity and acceleration, often acting at tangents to the circular path. The key characteristics of circular motion are:

• Uniform Motion: In a perfect circle, every point on the path maintains a constant speed.
• Centripetal Force: There is always a force directed towards the center of the circle, which maintains the circular nature of the motion.

Though Earth's actual path is slightly elliptical, assuming a circular path allows us to employ straightforward calculations like circumference to estimate distances traveled.

Distance Traveled

To find the distance traveled by Earth in one day, first, we need to know how far it travels in a year.
Using the circumference of Earth's orbit, which we've calculated as 584 million miles, we can determine the daily travel distance by dividing this annual travel distance by 365 days.This is computed as follows:

• Total annual travel distance \( = 584 \times 10^6 \text{ miles} \)
• Distance traveled per day \( = \frac{584 \times 10^6}{365} \approx 1.6 \times 10^6 \text{ miles} \)

Thus, the Earth travels roughly 1.6 million miles each day along its orbit. This calculation allows us to comprehend the immense speed and vast distance involved in Earth's daily journey around the Sun.