Problem 71 - Precalculus

Question

Find the linear speed of the moon if the average distance between the earth and moon is \(239,000\) miles, assuming the orbit of the moon is circular and requires about 28 days. Express answer in miles per hour.

Step-by-Step Solution

Answer

The linear speed of the moon is approximately 2,235.93 miles per hour.

Step 1: Determine Orbit Circumference

The first step is to calculate the circumference of the orbit of the moon. Since the orbit is circular, the formula for circumference is given by \( C = 2\pi r \), where \( r \) is the radius of the orbit. Here, \( r = 239,000 \) miles. Thus, the circumference \( C \) is \( 2 \times \pi \times 239,000 \approx 1,501,936 \) miles.

Step 2: Convert Orbital Period to Hours

Next, convert the time it takes for the moon to complete one orbit from days to hours. Since the moon's orbital period is 28 days and there are 24 hours in a day, the period in hours is \( 28 \times 24 = 672 \) hours.

Step 3: Calculate Linear Speed

Finally, the linear speed \( v \) of the moon can be found by dividing the total distance traveled (the circumference of its orbit) by the time taken to travel that distance. The linear speed is \( v = \frac{C}{ ext{time}} = \frac{1,501,936}{672} \approx 2,235.93 \) miles per hour.

Key Concepts

Circular Orbit

When the exercise refers to a 'circular orbit,' it means that the path traveled by the moon as it revolves around the Earth forms a perfect circle. In real life, the moon's orbit is slightly elliptical, but for simplification here, we assume a circular path.
When an object is in a circular orbit, the consistent distance from the center of the circle to any point on the orbit is known as the radius. This radius is crucial in calculating other aspects such as the circumference of the orbit. In the exercise given, the average distance, or the radius of the moon’s circular orbit, is specified as 239,000 miles.

• The concept of a circular orbit helps simplify complex celestial mechanics into manageable calculations.
• It assumes uniform distance from the central body through the course of the orbit.

Understanding the notion of a circular orbit is foundational in celestial mechanics, aiding in the calculations of linear speed and many other properties of orbital motion.

Orbit Circumference

In dealing with objects in circular orbits, the circumference is the total distance traveled in one complete rotation around the orbit. It's like measuring the perimeter of a circle. The calculation of the orbit's circumference is crucial because it directly impacts the determination of linear speed.
For a circular orbit, the formula to find the circumference is \( C = 2\pi r \). In this formula, \( r \) is the radius of the orbit. By using \( \pi \approx 3.14159 \), you can calculate the circumference.

• For example, with a radius of 239,000 miles, the circumference becomes \( 2 \times 3.14159 \times 239,000 \approx 1,501,936 \) miles. This number represents the complete path the moon travels around the Earth.

Calculating the orbit circumference is a significant step when analyzing any orbital motion since it lays the groundwork for understanding how far an object travels in its orbit.

Orbital Period

The 'orbital period' is the time it takes for the moon to make one full trip around the Earth. In this exercise, the moon completes its orbit in approximately 28 days. Knowing this period helps in computing the linear speed of the moon.
To convert the orbital period into hours (since speed often requires time in hours), multiply the number of days by 24, the number of hours in a day. That gives you the total period in hours.

• The orbital period of the moon being 28 days means \( 28 \times 24 = 672 \) hours.

This is important because, when you pair the circumference with the orbital period in hours, you can compute how fast the moon travels in terms of miles per hour. The orbital period is a key variable in determining the linear speed, emphasizing the relationship between distance traveled and time taken.