How far above the earth's surface is the satellite orbit?

In summary, the question asks about the distance above the Earth's surface that two satellites' orbits are at. The first satellite passes overhead at the equator and then again 15.0 hours later, while the second satellite passes overhead and then again after 15.0 hours. The equation T2=4π2r3/GM is used to solve for the radius of the orbit, accounting for the rotation of the Earth. By drawing a picture and using proportions, the correct distance of the satellite from the Earth's surface is found.
  • #1
Julie323
24
0

Homework Statement



A. Suppose you are at the Earth's equator and observe a satellite passing directly overhead and moving from west to east in the sky. Exactly 15.0 hours later, you again observe this satellite to be directly overhead. How far above the Earth's surface is the satellite's orbit?

B. You observe another satellite directly overhead and traveling east to west. This satellite is again overhead in 15.0 hours. How far is this satellite's orbit above the surface of the earth?



Homework Equations



T2=4π2r3/GM

The Attempt at a Solution


I was not sure how to account for the rotation of the earth, but I tried 15/(1+15/24)=9.23 hours, and then used that for T. I plugged in 6.67*10-11 for G and 5.97 *1024 for M. I solved for r giving me 2.23*107 meters.

Where did I go wrong? Thanks so much for any help!
 
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  • #2
Suggest you draw a picture of the Earth and show the 15/24th turn the Earth makes. And the turn that the satellite makes going the other way round (for part a).

After you find how far round it goes in 15 hours, use a proportion to find its time to go a full orbit.
 
  • #3
You got the radius of the orbit. The question is the distance of the satellite from the surface of Earth.

ehild
 
  • #4
O right! I forgot to subtract the radius of the earth. I got both parts, thank you so much!
 
  • #5



I would like to offer a different approach to solving this problem. Instead of using the given time intervals and trying to account for the rotation of the Earth, we can use the formula for the period of an orbiting object to calculate the height of the satellite's orbit.

T = 2π√(a3/GM)

Where T is the period of the orbit, a is the semi-major axis (half the length of the longest diameter of the elliptical orbit), G is the gravitational constant, and M is the mass of the Earth.

For part A, we can use the given information to calculate the period of the satellite's orbit:

T = 15 hours * 3600 seconds/hour = 54,000 seconds

Using this value for T and plugging in the values for G and M, we can solve for a:

54000 = 2π√(a3/(6.67*10^-11 * 5.97 * 10^24)

Solving for a gives us a value of 4.23*10^7 meters.

Since the semi-major axis is half the length of the longest diameter of the orbit, the distance from the center of the Earth to the satellite's orbit is half of this value, or 2.12*10^7 meters.

For part B, the process is the same. Using the given information, we can calculate the period of the satellite's orbit to be 54,000 seconds. Plugging this into the formula and solving for a gives us a value of 2.12*10^7 meters for the semi-major axis. Therefore, the distance from the center of the Earth to the satellite's orbit is 1.06*10^7 meters.

In conclusion, the distance from the center of the Earth to the satellite's orbit is 2.12*10^7 meters for part A and 1.06*10^7 meters for part B. This means that the satellite's orbit is approximately 2.12*10^7 meters and 1.06*10^7 meters above the Earth's surface for parts A and B, respectively.
 

1. What is the average distance between a satellite and the earth's surface?

The average distance between a satellite and the earth's surface is approximately 22,236 miles or 35,786 kilometers.

2. How high above the earth's surface is a geostationary satellite?

A geostationary satellite is typically placed at an altitude of about 22,236 miles or 35,786 kilometers above the earth's surface. This allows it to orbit at the same speed as the earth's rotation, making it appear stationary from the ground.

3. How far above the earth's surface is a polar orbiting satellite?

A polar orbiting satellite is usually placed at an altitude of around 500 miles or 800 kilometers above the earth's surface. This allows it to pass over both the north and south poles on each orbit.

4. Does the distance between a satellite and the earth's surface vary?

Yes, the distance between a satellite and the earth's surface can vary depending on factors such as the type of orbit, altitude adjustments, and atmospheric drag.

5. How do scientists calculate the distance between a satellite and the earth's surface?

Scientists use mathematical equations and measurements of the satellite's orbit to calculate its distance from the earth's surface. This can also be done using specialized tracking equipment and ground stations.

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