Positional Probability of Periodic Object Motion

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SUMMARY

The discussion focuses on calculating the positional probability of an object in periodic motion using its velocity as a function of position. The example provided utilizes the equations X(t) = A sin(wt) and V(t) = Aw cos(wt) to derive the probability density function P(X) through a change of variables. The final formula for positional probability is P(X) = (w/(2π))/(Aw(1-(X/A)²)^(1/2)). Participants emphasize the importance of avoiding divide by zero errors when X approaches ±A.

PREREQUISITES
  • Understanding of periodic motion and sinusoidal functions
  • Familiarity with calculus concepts, particularly change of variables
  • Knowledge of probability density functions
  • Basic understanding of trigonometric identities
NEXT STEPS
  • Study the derivation of probability density functions in periodic systems
  • Learn about change of variables in calculus
  • Explore the implications of divide by zero errors in mathematical modeling
  • Investigate applications of positional probability in physics and engineering
USEFUL FOR

Physicists, mathematicians, and engineers interested in the dynamics of periodic motion and probability theory will benefit from this discussion.

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If you know the velocity of an object as a function of position Can you use a uniform distribution over one period and the object velocity to perform a change of variables for the positional probability.

Example.
X(t)=Asin(wt)
V(t)=Awcos(wt)
V(X)=+-Aw(1-(X/A)^2)^(1/2)
P(t)=1/T
T=Period

Change of Variables
P(X)=P(t)|dt/dX|=P(t)V^-1=
(w/(2pi))/(Aw(1-(X/A)^2)^(1/2)

It looks like you could do this over any time interval t=(a,b) if you know the velocity as a function of position.
 
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You will probably want to be careful of your divide by zero errors when x = +/- a.
 

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