Proper time of an accelerating particle

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Discussion Overview

The discussion revolves around calculating the proper time experienced by a particle undergoing constant acceleration from rest to 0.5c over a duration of 3 seconds, as measured in a laboratory frame. The conversation includes theoretical considerations and mathematical approaches to the problem.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about the proper time for a particle with constant acceleration, expressing uncertainty about how to approach the problem.
  • Another participant suggests determining the world line of the particle and mentions the equation d²x/dt² = a, indicating that integrating this could yield the position as a function of time.
  • A different participant seeks clarification on how to derive proper time from the position function and confirms that they are referring to constant acceleration, not proper acceleration.
  • Another response emphasizes the importance of knowing the velocity as a function of time to define proper time and suggests using boundary conditions to determine necessary constants in the equations.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and approaches to the problem, indicating that there is no consensus on the best method to calculate the proper time. The discussion remains unresolved with multiple perspectives presented.

Contextual Notes

Participants mention the need for specific boundary conditions and the choice of coordinate system, which may affect the constants in their equations. There is also a distinction made between constant acceleration and proper acceleration, which could influence the calculations.

alienslag
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A particle has a constant acceleration in a laboratory from 0 to 0.5c in 3 seconds. What time elapses for the particle (i.e. what is the proper time for the particle)? Hint: (you will have to integrate the proper time of the particle over the 3 seconds as measured in the laboratory frame).

Not sure how to approach this problem. Any help would be greatly appreciated.
 
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You have to figure out what the world line looks like. You know that [itex]d^2x/dt^2=a[/itex], where a is a constant and you can integrate that equation to find x as a function of t.

If by any chance you meant "constant proper acceleration" when you said "constant acceleration", it's more difficult to figure out what the world line looks like, but DrGreg derived it here, starting with post #13 and ending with a correction in post #28.
 
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I am not sure I follow, could you please elaborate. X as a function of t is 1/2at^2 + volt +xo, but then how would I find the proper time of the particle from this relationship? Also I mean constant acceleration and not proper acceleration.
 
Do you know the definition of proper time? When you write it down, it's clear that you need to know the velocity as a function of time. (Position as a function of time is actually less useful). So what you need to do now is to find out the values of the constants in the equation you found. The boundary conditions that you stated in post #1 contain the information you need for that. The values of the constants depend on where you choose to put the origin of your coordinate system, so try to make a choice that simplifies the math as much as possible.
 

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