Accurate position change for accelerating objects

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SUMMARY

This discussion focuses on accurately calculating the position change of an object under unpredictable forces using the formula x_{2}=\frac{1}{2}at^{2}+vt+x_{1}. The challenge presented is to achieve consistent position values after varying time intervals, specifically comparing a single 2 ms interval to two 1 ms intervals. The solution requires a model for the acceleration profile, allowing for the use of average acceleration over small timesteps to maintain accuracy across different intervals.

PREREQUISITES
  • Understanding of kinematics, specifically the equations of motion.
  • Familiarity with instantaneous acceleration and velocity concepts.
  • Knowledge of numerical methods for approximating solutions in physics.
  • Basic experience with modeling and simulation techniques.
NEXT STEPS
  • Research numerical integration techniques for solving differential equations in physics.
  • Explore methods for modeling unpredictable forces in dynamic systems.
  • Learn about averaging techniques for acceleration over varying time intervals.
  • Investigate the use of Runge-Kutta methods for improved accuracy in position calculations.
USEFUL FOR

Physicists, engineers, and computer scientists involved in motion simulation, particularly those working with dynamic systems under variable forces.

Droctagonopus
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If an object is subject to different forces at different times and these forces are totally unpredictable (the force at any instant after the current time cannot be predicted). How would we make the position change due to acceleration as accurate as possible?

I have chosen a method but I've encountered a problem.
Use x_{2}=\frac{1}{2}at^{2}+vt+x_{1} where a is the instantaneous acceleration, v is the instantaneous velocity, x_{1} is the position at the current time and t is a very small time interval. However, I need a way to find the same position value after unequal time intervals. Meaning that if I take a single 2 ms interval in one case and two 1 ms intervals in another, the final value doesn't have to be too accurate but it has to be the same for both cases. Is there an efficient way to do this?
 
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You will need some model how your accelerations look like. Different step sizes are not an issue with your formula, if you know an average acceleration within (small) timesteps.
 

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