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If I have a rocket constantly accelerating at some rate a, (requiring some constant force F), I want to know how much work is done by the rocket to travel a given proper length.
For example, a rocket traveling to alpha centauri at 1g acceleration half way and 1g acceleration the other half. It would seem work would just be F*d, however, the d should be as measured in the rocket's frame of reference. Would a correct method of deriving the d to be to take the velocity function of the rocket, integrate from 0 to some t (half way point), and then multiply by two (symmetry) to obtain total distance traveled in the frame of the rocket?
General note: the function I'm using for the time between two points also requires the measurement d. The equation is:
t=\frac{c}{a_0} Cosh^{-1}\left[1 + \frac{a_0 d}{c^2}\right]
with t being as measured on the ship. To get the time to the mid-way point (required for the above approach), is it valid to just set d/2? (and also multiply by two to get the entire shipboard time for the full voyage)
For example, a rocket traveling to alpha centauri at 1g acceleration half way and 1g acceleration the other half. It would seem work would just be F*d, however, the d should be as measured in the rocket's frame of reference. Would a correct method of deriving the d to be to take the velocity function of the rocket, integrate from 0 to some t (half way point), and then multiply by two (symmetry) to obtain total distance traveled in the frame of the rocket?
General note: the function I'm using for the time between two points also requires the measurement d. The equation is:
t=\frac{c}{a_0} Cosh^{-1}\left[1 + \frac{a_0 d}{c^2}\right]
with t being as measured on the ship. To get the time to the mid-way point (required for the above approach), is it valid to just set d/2? (and also multiply by two to get the entire shipboard time for the full voyage)