How Old is Pollux When He Returns to Earth?

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SUMMARY

The discussion centers on a relativistic physics problem involving identical twins, Castor and Pollux, where Pollux travels in a rocket ship that accelerates and decelerates over a series of 5-year intervals. The key equations involve relativistic velocity and time dilation formulas, which are essential for determining Pollux's age upon return, the year on Earth, and the distance traveled. The problem requires an understanding of how to calculate motion in a non-inertial frame and the implications of changing gamma factors in time dilation. The user, Daniel, seeks assistance in deriving the equations of motion for the rocket from the perspective of the astronauts.

PREREQUISITES
  • Understanding of relativistic velocity formulas
  • Familiarity with time dilation concepts in special relativity
  • Knowledge of acceleration in non-inertial reference frames
  • Ability to apply calculus to derive motion equations
NEXT STEPS
  • Study the derivation of the relativistic rocket equations of motion
  • Learn about the effects of acceleration on time dilation in special relativity
  • Explore the concept of proper time and its calculation in non-inertial frames
  • Investigate the implications of changing gamma factors in relativistic physics
USEFUL FOR

Students and enthusiasts of physics, particularly those interested in special relativity, time dilation, and the twin paradox scenario. This discussion is beneficial for anyone looking to deepen their understanding of relativistic motion and its mathematical foundations.

dwintz02
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Homework Statement


Assume a rocket ship leaves the Earth in the year 2100. Castor, one of a set of identical twins born in 2080, remains on Earth to work at Mission Control, while the other twin, Pollux, travels in the rocket. Ignore the motion of the Earth relative to the fixed stars. The rocket is constructed so that it has an acceleration g in its instantaneous rest frame (making the astronauts feel at home) Suppose the rocket accelerates in a straight line path for 5 years (by its OWN chronometer), decelerates at the same rate for another 5 years (of its own time), turns around (in negligible time), accelerates back for 5 years, decelerates for 5 years, and then lands on earth.

a) How old is Pollux according to his watch?
b) What year is it on Earth?
c) How far away from the Earth did the rocket travel?


Homework Equations



Relativistic velocity formulas, time formulas, and acceleration formulas.



The Attempt at a Solution



So I've taken a time derivative of the velocity formulas and it matches with my book, but this only gives me accelerations in terms of other reference frames. What I'm having trouble with is calculating the equation of motion for the rocket according to the astronauts. How can I continuously add their velocities (because they are accelerating), to say get their velocity as a function of time in their frame? And once I get that, how can I handle the fact that gamma is changing in the time dilation formula?

Thank you in advance,

Daniel
 
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