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I am stationary and I want to travel to a destination that is 1 light-year away. This journey needs to last 1 year for me. What is the average speed that my rocket must travel to achieve this?

Consider the stationary reference frame for the distance and the moving reference frame for the time.

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    $\begingroup$ What conceptual questions do you have that would help you solve this homework-like problem? $\endgroup$
    – Ghoster
    Commented Feb 21, 2023 at 3:53
  • $\begingroup$ a distance of 1 light-year Measured by whom? $\endgroup$
    – Ghoster
    Commented Feb 21, 2023 at 3:55
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    $\begingroup$ @Ghoster, the distance is measured in the stationary reference frame and the time in the moving reference frame. $\endgroup$
    – Eduardo M
    Commented Feb 21, 2023 at 5:04
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    $\begingroup$ @EduardoM No, because acceleration in the frame of the Earth won't be constant. $\endgroup$
    – Ján Lalinský
    Commented Feb 21, 2023 at 7:20
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    $\begingroup$ @EduardoM Accelerating for half a year and decelerating for the other half with the right acceleration to make 1 light year in 1 year of proper time is a different motion from the uniform motion with the right velocity to do the same. It changes how long the trip takes in Earth's frame, due to continuously changing Lorentz factor in the law of time dilation. $\endgroup$
    – Ján Lalinský
    Commented Feb 23, 2023 at 17:23

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The proper time is $$\tau^2=t^2-x^2$$ in units where $c=1$. The distance is $x=1$ and the coordinate time is $t=1/v$. So the proper time is $$1^2=\left(\frac{1}{v}\right)^2 - 1^2$$$$v=\frac{1}{\sqrt{2}}$$

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