Odds of Collision between Incident Object and Target Objects

[thepopasmurf]

Homework Statement

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.

Homework Equations

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

σ = $\pi$ (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

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

 

[mfb]

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.