Probability of cosine of angle between two directions in collision

[albertrichardf]

The question refers to the Feynman lectures on physics Vol I chapter 39. He discusses collisions between gas molecules. Here is a relevant extract:

They are equally likely to go in all directions, but how do we say that? There is of course no likelihood that they will go in any specific direction, because a specific direction is too exact, so we have to talk about per unit “something.” The idea is that any area on a sphere centered at a collision point will have just as many molecules going through it as go through any other equal area on the sphere. So the result of the collisions will be to distribute the directions so that equal areas on a sphere will have equal probabilities.

Incidentally, if we just want to discuss the original direction and some other direction an angle ø from it, it is an interesting property that the differential area of a sphere of unit radius is sin ø dø times 2π. And sin ø dø is the same as the differential of - cos ø. So what it means is that the cosine of the angle ø between any two directions is equally likely to be anything from -1 to 1.

My question is how does he conclude that cos ø could be anything from 1 to -1 based on the idea that equal areas have an equal number of molecules passing through? I can't see that at all. The first paragraph just compares areas, but when he talks about cos ø he puts forth only one area: That between the two directions. So how does he go from 2 areas to one only?

Here is the link to the chapter: http://www.feynmanlectures.caltech.edu/I_39.html It is in section four.

Thanks for any answers

 

[pixel]

There is an original direction, vertically upward in his Fig. 32-1, and a new direction specified by the polar angle θ. He calculates a differential annular segment with area 2π sinθ dθ (that includes all azimuthal angles as these are equally probable by symmetry). All such annular segments will have equal probabilities to contain molecules, that is, equal numbers of molecules going through it as through any other annulus. Now it's just a mathematical fact that d(-cosθ) = sinθ dθ so any value of cosθ is equally likely to contain molecules. (It's a bit confusing that he is using ∅ in the paragraph and θ in the figure).

 

[albertrichardf]

There is an original direction, vertically upward in his Fig. 32-1, and a new direction specified by the polar angle θ. He calculates a differential annular segment with area 2π sinθ dθ (that includes all azimuthal angles as these are equally probable by symmetry). All such annular segments will have equal probabilities to contain molecules, that is, equal numbers of molecules going through it as through any other annulus. Now it's just a mathematical fact that d(-cosθ) = sinθ dθ so any value of cosθ is equally likely to contain molecules. (It's a bit confusing that he is using ∅ in the paragraph and θ in the figure).

Oh alright. So essentially the cos theta comes from the coordinate system rather than the angle forming the area. Thanks for the explanation.

 

[pixel]

Oh alright. So essentially the cos theta comes from the coordinate system rather than the angle forming the area. Thanks for the explanation.

I'm not sure what you mean by this.

 

[Physics Student95]

There is an original direction, vertically upward in his Fig. 32-1, and a new direction specified by the polar angle θ. He calculates a differential annular segment with area 2π sinθ dθ (that includes all azimuthal angles as these are equally probable by symmetry). All such annular segments will have equal probabilities to contain molecules, that is, equal numbers of molecules going through it as through any other annulus. Now it's just a mathematical fact that d(-cosθ) = sinθ dθ so any value of cosθ is equally likely to contain molecules. (It's a bit confusing that he is using ∅ in the paragraph and θ in the figure).

Can you please explain why you say any value of cosθ is equally likely to contain molecules (Feynman said this by saying cosθ is equally likely to be anything from -1 to +1). You mentioned that d(-cosθ) = sinθ dθ, but I don't see why this means cosθ is equally likely to be anything from -1 to +1.