Will Two Synchronized Atomic Clocks Remain in Sync After One Year?

[Lauren12]

Homework Statement

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

Δtm = Δts/√(1-v^2/c^2)

The Attempt at a Solution

No, the two clocks will not be synchronized after one year. The clock on the satellite will lose very small amount of time due to their motion with respect to the clocks on earth. The clock in outer space has changed frame of reference while the clock on Earth has not.

b)
If the speed of light was only twice the average speed of the satellite more time dilation would occur. As objects approach the speed of light, time becomes dilated.
Let x= the speed of the satellite
therefore 2x= the speed of light
Is it a valid move to call it x? They do not give us a velocity for the satellite...
t_s= 3.1536 * 10^7 (seconds in a year)

Δtm = Δts/√(1-v^2/c^2)
Δtm = Δts/√(1-(x^2)/(4x^2))
Δtm =(3.1536 * 10^7) /√(1-0.25)
Δtm =36414636.18

Since they said "explain your reason with calculations" do you think it's necessary to do a calculation for a) ?
Thanks for any input! :D

 

[Tobbin92]

If the question was given in the same way as above then I suspect that you only would have to justify your answer with a calculation in b). But my experience says that a calculation as extra justification will never harm.

Therefore you might add a mathematical argument to support your answer in question a) if you feel unsure.