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At what speed would one day = one year?

  1. Dec 11, 2011 #1
    For instance at what speed would something have to be going relative to the Earth that only one day happens for this FOR but 1 years passes for the Earth?

    What speed would a spaceship have to be going where it experiences one full day while 1 year on earth has passed?

    The guy on the spaceship experiences one full day but a years time has elapsed on Earth. What speed is the spaceship going?
     
  2. jcsd
  3. Dec 11, 2011 #2
    365sqrt{1-v2/9*1016}=1
    solve for v
     
  4. Dec 11, 2011 #3

    ghwellsjr

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    The time dilation factor, gamma, or γ, is a function of β, the speed as a fraction of the speed of light. The formula is:

    γ = 1/√(1-β2)

    What you want is the formula for β as a function of γ, so we can rearrange the equation as follows:

    γ = 1/√(1-β2)
    γ2 = 1/(1-β2)
    1-β2 = 1/γ2
    -1+β2 = -1/γ2
    β2 = 1-1/γ2
    β = √(1-1/γ2)

    So if you take a year to be equal to 365.25 days, then γ=365.25, so we plug it in and turn the crank:

    β = √(1-1/γ2)
    β = √(1-1/365.252)
    β = √(1-1/133407.5625)
    β = √(1-0.0000074958)
    β = √(0.9999925042)
    β = 0.999996252

    As a sanity check, we can plug this value of β into the formula for gamma and see that we get 365.25:

    γ = 1/√(1-β2)
    γ = 1/√(1-0.9999962522)
    γ = 1/√(1-0.9999925042)
    γ = 1/√(0.0000074958)
    γ = 1/(0.00273785)
    γ = 365.25
     
    Last edited: Dec 11, 2011
  5. Dec 11, 2011 #4
    So if someone is traveling at 0.999996252c relative to the Earth, one complete day will pass for them while one year passes for someone on Earth?
     
  6. Dec 11, 2011 #5

    ghwellsjr

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    In a frame in which the Earth is at rest, one year will pass on the Earth, while 1 day passes on the spaceship traveling at that speed.

    But, for the same scenario, in a frame in which the spaceship is at rest, one year will pass on the spaceship, while 1 day passes on Earth.

    Or to put it another way, an observer on the Earth will measure the clocks on the spaceship to be ticking at 1/365.25 of the rate of a clock on Earth and an observer on the spaceship will measure the clocks on Earth ticking at 1/365.25 of the rate of a clock on the spaceship. But they will make this measurement indirectly as expressed by the Relativistic Doppler factor which is exactly symmetrical. This is the ratio of the rate that the relatively moving clock is ticking compared to their own clock and is not the same factor as gamma. The Relativistic Doppler describes what each observer actually sees and is not dependent on any frame or theory of relativity whereas time dilation is.
     
  7. Dec 11, 2011 #6
    Let's say the the person who travels at .99c and experiences this one day comes back to Earth.

    Does he come back to earth and one full year has passed on Earth?
     
  8. Dec 11, 2011 #7

    ghwellsjr

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    If he traveled at the speed we calculated earlier (much, much faster then .99c), then, yes, when he returned after experiencing a one-day trip, it would be a year later, which is a good thing because if he came back to the same place he left at some other speed, the earth might not be there when he got back.
     
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