Rutherford Scattering mass of particle

[Rubiss]

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 T_min required if the two spheres are to have a chance of touching?

Homework Equations

$$cot(\frac{θ}{2}) = \frac{2Es}{ZZ^{'}e^{2}}$$

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=mv_os=s\sqrt(2mE) [/tex]

where l is the angular momentum, m is the mass, v_o 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 T_min). 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

 

[Rubiss]

Ok. Since I'm not receiving any help. Let's try again. Let me me start with a different problem (that I found in a problem book) for which I know the solution:

A particle of mass m is projected from infinity with a velocity v_o in a manner that it would pass a distance b from a fixed center of inverse-square repulsive force (magnitude k/r², where k is a constant) if it were not deflected. Find the distance of closest approach:

Solution:
When the particle is at the closest distance from the fixed center of force, $$\dot{r}=0.$$ Conservation of energy gives $$\frac{mv_{o}^{2}}{2} = \frac{k}{R}+\frac{mV^2}{2}$$ where R is the closest distance and $$V=R \dot{θ}$$ is the speed of the particle when it reaches the pericenter. Conservation of angular momentum gives $$v_{o}mb=mVR,$$ that is, $$V=\frac{v_{o}b}{R}.$$ Plugging this into our conservation of energy equation gives $$\frac{mv_{o}^{2}}{2} = \frac{k}{R} + \frac{mv_{o}^{2}b^{2}}{2R^{2}}$$ or $$R^{2} - \frac{2kR}{mv_{o}^{2}} - b^{2} = 0$$ which gives the closest distance of approach as $$R=\frac{k}{mv_{o}^{2}}+\sqrt{(\frac{k}{mv_{o}^{2}})^{2}+b^{2}}$$

Ok. Now let me try to apply this procedure to problem 2 in my first post above:

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?

When the particle is at the closest distance from the fixed center of force, $$\dot{r}=0.$$ Conservation of energy gives $$T = \frac{kQ_{1}Q_{2}}{R}+\frac{mV^2}{2}$$ where R is the closest distance and $$V=R \dot{θ}$$ is the speed of the particle when it reaches the pericenter. Conservation of angular momentum gives $$smv_{o}=RmV$$ where s is the impact parameter which is not given in the problem. We can rewrite the conservation of angular momentum equation as $$sm\sqrt{\frac{2T}{m}} = RmV$$ which tells us that $$V^2 = \frac{2Ts^{2}}{mR^{2}}.$$ Plugging this last expression into our conservation of energy equation gives $$T = \frac{s^{2}T}{R^{2}} + \frac{kQ_{1}Q_{2}}{R}$$ which after some algebra gives $$T = kQ_{1}Q_{2}\big(\frac{R}{R^{2}-s^{2}}\big).$$

Now I don't know where to go from here, and I don't know if I am headed in the right direction. From here I would need to say what the distance of closest approach (R) and the impact parameter (s) is in terms of the given R1 and R2. Can anyone please help?