Finding relativistic force in terms of acceleration

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SUMMARY

The discussion focuses on deriving the relativistic force in terms of acceleration using linear momentum in the context of special relativity. The key tools mentioned include Mathematica for computational assistance. The challenge lies in expressing the force in the S frame rather than the S' frame, necessitating the treatment of velocity as a function of time. Participants emphasize the need for a clear starting point to approach this derivation effectively.

PREREQUISITES
  • Understanding of relativistic mechanics and force transformations
  • Familiarity with linear momentum equations in special relativity
  • Proficiency in using Mathematica for mathematical computations
  • Knowledge of calculus, particularly differentiation with respect to time
NEXT STEPS
  • Study the derivation of relativistic momentum and its implications
  • Learn how to use Mathematica for solving physics equations
  • Explore the concept of force transformation between inertial frames
  • Investigate the relationship between velocity, acceleration, and time in relativistic contexts
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This discussion is beneficial for physics students, educators, and researchers focusing on relativistic dynamics and those seeking to enhance their computational skills in physics using Mathematica.

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Homework Statement


The instantaneous force F acting on a particle, as measured in frame S, is
png.latex?F%20%3D%20%5Cfrac%7Bd%5Cvec%7Bp%7D%7D%7Bdt%7D.png

Use the formula for the linear momentum (
D%3D%5Cfrac%7Bm%20%5Cvec%7Bu%7D%7D%7B%5Csqrt%7B1-%5Cfrac%7Bu%5E%7B2%7D%7D%7Bc%5E%7B2%7D%7D%7D%7D.png
) in and the definition of the acceleration a to show that
D%7D%7Bc%5E%7B2%7D%7Da_%7By%7D+%5Cfrac%7Bu_%7Bx%7Du_%7Bz%7D%7D%7Bc%5E%7B2%7D%7Da_%7Bz%7D%5D.png

D%7Bc%5E%7B2%7D%7D%29a_%7By%7D+%5Cfrac%7Bu_%7By%7Du_%7Bz%7D%7D%7Bc%5E%7B2%7D%7Da_%7Bz%7D%5D.png

s;%281-%5Cfrac%7Bu_%7Bz%7D%5E%7B2%7D+u_%7Bx%7D%5E%7B2%7D%7D%7Bc%5E%7B2%7D%7D%29a_%7Bz%7D%5D.png


The Attempt at a Solution


The professor said that this required use of programs such as mathematica in order ot solve, however, I can't figure out a way to solve this, I can derive the force transformation into another inertial reference frame, but the equation needs to be in terms of acceleration in the S frame, not the S' frame. I just need to know where to start with this.
 
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It looks like you will need to treat ## \mathbf{u} ## and its magnitude as functions of ## t ## and take the time derivative of the expression given for ## \mathbf{p} ##.
 
Geofleur said:
It looks like you will need to treat ## \mathbf{u} ## and its magnitude as functions of ## t ## and take the time derivative of the expression given for ## \mathbf{p} ##.
Is there anyway to do this without expressing u in terms of t? In the prevoud question I found the velocity transformation as follows:
c%5E%7B2%7D%7D.gif
 

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