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Odds of collision

  1. May 11, 2014 #1
    1. The problem statement, all variables and given/known data

    The problem is what are the odds of an incident object of radius r1 colliding with any of a collection of target objects of radius r2, where the r2 objects have a number density N / m^3 = n and the incident object travels a distance L. Incident object is moving much faster than the other objects so they can be considered still.


    2. Relevant equations

    Collision cross-section for collision between incident object and a single target is:

    σ = [itex]\pi[/itex] (r1 + r2)^2

    Probability of collision for a single target object is
    P1 = σ/A
    were A is the total area of the domain in question.

    Probability no collision for a single target object is 1-P1

    Maybe relevant, the mean free path is
    λ = 1 / nσ

    3. The attempt at a solution

    My thinking is, if probability of no collision for a single target is (1-p1), then if the incident object travels a distance L, the number of targets to consider is Ln. So the total probability for no collision is

    (1-p1)^(Ln)

    And probability of colliding with a single one of these is 1 minus this answer.

    Is this correct?

    I was also trying to use the mean free path but I wasn't sure how.

    Thanks
     
  2. jcsd
  3. May 11, 2014 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Your exponent has units, that needs another factor. And then you need the limit for an infinite area.

    The probability to have no collision before length L is an exponential distribution, where the mean free path gives the factor in the exponent. That is easier to set up.
     
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