Collision Mechanics and Differential Cross Section

[jayqubee]

Homework Statement

The problem concerns the collision of a particle on a rigid, fixed disk of radius R. The coefficient of friction is zero, and the coefficient of restitution is 0 < ε < 1, such that v_{perpendicular, final} = εv_{perpendicular, initial} .
A) Show that tanα_incident = εtanα_final (completed)
Where α is the angle from the velocity vector to the normal vector

B) Write a program to numerically solve for the differential cross section (main problem)

Homework Equations

impact parameter : b
Scattering angle : θ
differential cross section : dσ/dΩ = (b/sinθ)|db/dθ|

The Attempt at a Solution

I honestly do not know where to start here. I solved this problem for hard-sphere scattering but now solving for the impact parameter, b, leads to a dependence on α, which vanished in the hard-sphere case. Any tips on getting this started would be appreciated.

 

[mark726]

Thank you for your post regarding the collision of a particle on a rigid, fixed disk. I would like to offer some suggestions on how to approach this problem and write a program to numerically solve for the differential cross section.

Firstly, it is important to understand the physical principles involved in this problem. The coefficient of friction being zero indicates that the disk is perfectly smooth, and the coefficient of restitution being between 0 and 1 suggests that the collision is not perfectly elastic. This means that there will be some loss of energy during the collision, which will affect the final velocities of the particle.

Now, let's focus on part A of the problem, which asks to show that tanαincident = εtanαfinal. This relationship can be derived from the conservation of momentum and energy. Consider the particle approaching the disk with an initial velocity vperpendicular, initial and an angle αincident with the normal vector of the disk. After the collision, the particle will have a final velocity vperpendicular, final and an angle αfinal with the normal vector. From the conservation of momentum, we can write:

mvperpendicular, initial = mvperpendicular, final

And from the conservation of energy, we can write:

1/2mvperpendicular, initial^2 = 1/2mvperpendicular, final^2

Solving for vperpendicular, final and substituting into the equation for the conservation of momentum, we get:

tanαincident = vperpendicular, initial/vparallel, initial = vperpendicular, final/vparallel, final = tanαfinal

And since we know that vparallel remains constant during the collision, we can write:

tanαincident = εtanαfinal

Moving on to part B, to numerically solve for the differential cross section, we can use the Monte Carlo method. This involves generating a large number of random points within a certain range of the impact parameter b, and then calculating the corresponding scattering angle θ for each point. The differential cross section can then be approximated by taking the average of the values obtained for dσ/dΩ for each point. This process can be repeated multiple times to improve the accuracy of the result.

I hope this helps you get started on solving this problem. If you have any further questions or need clarification, please don't hesitate to ask. Good luck with your program!