I Positional Probability of Periodic Object Motion

AI Thread Summary
The discussion explores the use of a uniform distribution over one period of motion to determine positional probability when the velocity of an object is known as a function of position. It presents a mathematical framework for changing variables from time to position, highlighting the relationship between velocity and probability. The example provided illustrates how to derive the positional probability using the object's velocity and period. It emphasizes the importance of avoiding divide by zero errors at specific positions. This approach can be applied to any time interval if the velocity function is accurately defined.
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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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