# Altitude of Satellites above the Surface of the Earth

#### Bassa

1. The problem statement, all variables and given/known data

A) Suppose you are on Earth's equator and observe a satellite passing directly overhead and moving from west to east in the sky. Exactly 12.0 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 12.0 hours. How far is this satellite's orbit above the surface of the earth?

2. Relevant equations
T=(2πr^3/2)/(√Gm)

3. The attempt at a solution
Well, I thought that they would both have the same altitude since they have the same period. This is evidently not correct since the back of the book has two different answers for A and B.

I plugged in the numbers for the variables. I dropped the units. I would have looked very messy.

86400=(2πr^3/2)/(√(6.67x10^-11)(5.97x10^24))

I solved for r

r=42226910.15m

altitude= r-Re=35846910.18m which matched what the book has as an answer for B, but why is this the answer for B. Why would they both have different altitude if they have the same period, but in opposite direction? May I get any hint for part A and some clarification on part B?

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#### voko

What makes you think they have the same period?

#### Bassa

What makes you think they have the same period?
Well, from the description in the problem, I thought that they were both geosynchronous satellites. If so, would not that make them have the same period? The only difference would be that one is going in the direction of the earth's rotation and the other is going in the opposite direction.

#### voko

Well, from the description in the problem, I thought that they were both geosynchronous satellites.
Really? How is the geosynchronous orbit defined?

#### lep11

Why would they be geosynchronous? Clearly they are not geosyncronous because they are not stationary relative to a geographical location on the Earth.

#### lep11

the Earth is spinning counter-clockwise about its axis, so you must take that into account

#### Bassa

Really? How is the geosynchronous orbit defined?
A geosynchronous orbit is an orbit in which the satellite is locked over one place relative to the earth. The earth and the satellite would have the same orbital velocity. On a second thought, I don't see a lot of evidence to support the geosynchronous orbit hypothesis. They may have fast orbital velocity that would allow them to appear again on the same spot after a given time interval.

#### voko

You (the observer) are orbiting the Earth with angular speed one revolution per day. What angular speeds must the satellites have so that your angular positions and theirs overlap twice a day?

#### Jazz

Suppose you know a person is walking with velocity v. If the person and you are both moving toward each other, could you say that the person's absolute velocity is just v?

#### Bassa

Suppose you know a person is walking with velocity v. If the person and you are both moving toward each other, could you say that the person's absolute velocity is just v?
No, it would be his relative velocity to me.

#### Jazz

v would be the velocity relative to an observer at rest. It would be different if the observer is not so.

In the problem, you don't have a inertial frame of reference. You must figure out their absolute velocities.

If you had a geosynchronous satellite right over you, you would say that it's not moving at all, but that is not the case.

#### Bassa

v would be the velocity relative to an observer at rest. It would be different if the observer is not so.

In the problem, you don't have a inertial frame of reference. You must figure out their absolute velocities.

If you had a geosynchronous satellite right over you, you would say that it's not moving at all, but that is not the case.
What indicates to us that it is not geosynchronous?

#### mfb

Mentor
The satellites appear directly above the observer at the initial time and 12 hours later, but not in between. That rules out geostationary satellites (it would also make "move east to west" and "west to east" wrong).

How does the position of the observer on earth (relative to a non-rotating frame) change during 12 hours?

#### Bassa

The satellites appear directly above the observer at the initial time and 12 hours later, but not in between. That rules out geostationary satellites (it would also make "move east to west" and "west to east" wrong).

How does the position of the observer on earth (relative to a non-rotating frame) change during 12 hours?
So, after completing half a period on earth, the satellite appears again. Does that mean that the satellite completed 1.5 times the period of the earth?

#### Jazz

What indicates to us that it is not geosynchronous?
If I have the means to view a geosynchronous satellite, I would be able to view it whenever I want and I'd find it always in the same place in the sky.

If this doesn't happen, then it's not a geosynchronous satellite.

#### mfb

Mentor
So, after completing half a period on earth, the satellite appears again. Does that mean that the satellite completed 1.5 times the period of the earth?
What does "1.5 times the period of the earth" mean? The number 1.5 is correct for one of the satellites, I'm not sure if the interpretation is correct as well.

#### Bassa

What does "1.5 times the period of the earth" mean? The number 1.5 is correct for one of the satellites, I'm not sure if the interpretation is correct as well.
Sorry for not being very clear. I meant that the satellite must have circled the earth 1.5 times for it to satisfy the information of when it was observed the second time. When it was spotted the first time in situation A, it was not seen in the sky until 12 hours later. This means that the satellite passed the point at witch it was spotted the first time. It was heading toward the east then it came back to the original spot in the sky when it was first observed, but the observer was already gone due to the rotation of the earth. Then, when it continued going around the earth, it reached the point where the observer is at. This happened twelve hours later of when it was originally observed.

#### voko

So what is the period of the satellite that makes 1.5 revolutions when the Earth does 0.5?

#### Bassa

So what is the period of the satellite that makes 1.5 revolutions when the Earth does 0.5?
according to my calculations it should be 8.0 hours or 28800 seconds. This is for situation A.

#### Bassa

I think I am going to finish part A first and then move to part B. ^-^

#### Bassa

My next move is to use T=2pi(r^3/2)/sqrt(GMe)
Me= mass of the earth
T= period of the satellite
Am I in the right direction?

#### voko

Yes, that seem OK.

#### Bassa

Now, for B its period is the same as the period of the earth. This mean that the same method will apply, but I would have to use 24 86400 seconds for the period in my formula. Does that sound okay? I used a solar system mobile that my girlfriend got me so I could wrap my head around it. I wonder if I am allowed to take the mobile with to a test. ^_^

#### voko

Same period as the period of the Earth is correct, but what is 24 86400 seconds?

You do not need a mobile to understand this. Take your girlfriend instead, draw a circle in the ground, and have fun.

"Altitude of Satellites above the Surface of the Earth"

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