How did my friend get this "Theoretical Velocity" ?

[AlexisFair]

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

Two atomic clocks are synchronized. One is placed on a satellite which orbits around the Earth at high speeds for a whole year. The other is placed in a lab and remains at rest with respect to the earth. You may assume both clocks can measure time accurately to many significant digits.

a) Will the two clocks still be synchronized after one year?

b) imagine the speed of light is much lower than its actual value. How would the results of this experiment change if the speed of light was only twice the average speed of the satellite? Explain your reasoning using a calculation.

Homework Equations

The Attempt at a Solution

Someone already posted this homework question (See link below) and I was wondering if someone could answer a few questions.

So for question A
The person the forum, found the theoretical time in seconds for one year, which is easy enough to get however, how did this person find theoretical velocity (3x103m/s) ? And is that velocity based on the atomic clock in respect to the Earth's reference? Question B)

Average speed of a satellite, is that something that can vary? IF so how would I know which speed to use?
Sorry this question is very confusing for me.

Reference:

https://www.physicsforums.com/threads/atomic-clock-time-dilation-experiment.565750/

 

[mjc123]

3000 m/s is approximately the speed of a satellite in geostationary orbit: https://en.wikipedia.org/wiki/Geostationary_orbit. In low Earth orbit it would be about 8000 m/s.

Average speed of a satellite, is that something that can vary?

Not really, if it keeps the same orbit. Average speed will vary with orbital height.

 

[Janus]

Since nothing is stated in the problem regarding the altitude of the orbiting clock (which also determines v), it seems he simply chose what he felt was a convenient value. The problem with the value he chose was that it put the orbiting clock at a higher altitude than the Earth surface clock which meant that he also had to take gravitational time dilation into account. He could have made it easier for himself by assuming that the clock orbited just above the surface of the Earth at the same height as the other clock. Then by using the equation for orbital velocity, he could have worked out a general equation that would have helped with part b without his ever having to know the actual value of v.

 

[AlexisFair]

Thank you guys for the help! When I saw 3 X 10^3 I didn't know what it was and thought maybe it was a set value that i forgot to write in my notes.

Also, would you guys happen to know how the person got the value "1x10^-6" for question B?

I tried to solving for "c" in the equation Δtm = Δts/√(1-v2/c2) and I got c= 3,000,000 and that didn't look right to me since the question says " How would the results of this experiment change if the speed of light was only twice the average speed of the satellite?" I mean shouldn't "c" be 6000 m/s since 2(3000) is twice the average speed of a satellite?

 

[AlexisFair]

Since nothing is stated in the problem regarding the altitude of the orbiting clock (which also determines v), it seems he simply chose what he felt was a convenient value. The problem with the value he chose was that it put the orbiting clock at a higher altitude than the Earth surface clock which meant that he also had to take gravitational time dilation into account. He could have made it easier for himself by assuming that the clock orbited just above the surface of the Earth at the same height as the other clock. Then by using the equation for orbital velocity, he could have worked out a general equation that would have helped with part b without his ever having to know the actual value of v.

3000 m/s is approximately the speed of a satellite in geostationary orbit: https://en.wikipedia.org/wiki/Geostationary_orbit. In low Earth orbit it would be about 8000 m/s.

Not really, if it keeps the same orbit. Average speed will vary with orbital height.

Please let me know if you understand what I mean. for question B. I would like to understand it a bit more.

 

[Janus]

Please let me know if you understand what I mean. for question B. I would like to understand it a bit more.

Question b is quite straight forward if you think about it. It basically breaks down to this:
What happens to the time dilation formula if you substitute 2v for c? (note that you don't actually have to know the value of v to do this.)

 

[AlexisFair]

Question b is quite straight forward if you think about it. It basically breaks down to this:
What happens to the time dilation formula if you substitute 2v for c? (note that you don't actually have to know the value of v to do this.)

Δtm = Δts/√(1-v²/c²)
Δtm = Δts/√(1-v²/(2v)²)
Δtm = Δts/√(1-(1/4))
Δtm = 3.1x10⁷/√(1-(1/4))
Δtm = 35795716.69

Correct?

If this is correct, then it supports the fact that time dilation happens a lot more when matter approaches the speed of light. In this situation since speed of light is reduced to twice the average speed of a satellite, time dilation is less. Unless I did something wrong of course...