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Homework Statement
I'm taking a graduate level course in classical mechanics that uses Goldstein's book. We are currently discussing scattering in a central field in chapter 3. Here are two problems that might be very basic/standard scattering problems, yet I'm not how to proceed or get started.
I'm assuming I need to use all the machinery of Rutherford scattering in chapter 3 of Goldstein rather than basic conservation of energy and momentum. Then again, I could be wrong.
Problem 1:
An alpha particle scatters directly backward after colliding with a nucleus of unknown mass. After the scattering event, the alpha particle loses 64% of its energy. Assume the mass of the alpha particle is 4 atomic mass units and that the scattering is elastic. Calculate the mass of the unknown nucleus.
Problem 2:
Sphere 1 has a mass m1, radius R1, and charge Q1. Sphere 2 has a mass m2, radius R2, and charge Q2. Sphere 2 is shot with kinetic energy toward sphere 1, which is initially at rest. What is the minimum kinetic energy Tmin required if the two spheres are to have a chance of touching?
Homework Equations
[tex] cot(\frac{θ}{2}) = \frac{2Es}{ZZ^{'}e^{2}} [/tex]
where θ is the scattering angle, E is the energy of the incoming particle, s is the impact parameter, Z is the atomic number of one of the particles, Z' is the atomic number of the other particle, and e is the elementary electric charge.
[tex] l=mvos=s\sqrt(2mE) [/tex]
where l is the angular momentum, m is the mass, vo is the initial velocity, s is the impact parameter, and E is the energy.
We are also talking about transformation of coordinates between the lab and center and mass frame, and I'm not sure if that would be of use in these problems.
The Attempt at a Solution
For problem 1, we know the scattering angle (180 degrees), we also know the charge of the alpha particle. I don't know what else I could get from the equation.
In problem 2, we know the impact parameter must be less than R1+R2, and we know the energy (it is Tmin). I don't know where to go from there.
As I said earlier, I think it would be easy to solve these problems using standard first-year conservation of energy and momentum, but I assume I need to use upper-level/graduate ideas to solve the problems. Any help would be appreciated. Thanks