• Central Forces 2023

    Consider a very light particle of mass \(\mu\) scattering from a very heavy, stationary particle of mass \(M\). The force between the two particles is a repulsive Coulomb force \(\frac{k}{r^2}\). The impact parameter \(b\) in a scattering problem is defined to be the distance which would be the closest approach if there were no interaction (See Figure). The initial velocity (far from the scattering event) of the mass \(\mu\) is \(\vec v_0\). Answer the following questions about this situation in terms of \(k\), \(M\), \(\mu\), \(\vec v_0\), and \(b\). (It is not necessarily wise to answer these questions in order.)

    1. What is the initial angular momentum of the system?
    2. What is the initial total energy of the system?
    3. What is the distance of closest approach \(r_{\rm{min}}\) with the interaction?
    4. Sketch the effective potential.
    5. What is the angular momentum at \(r_{\rm{min}}\)?
    6. What is the total energy of the system at \(r_{\rm{min}}\)?
    7. What is the radial component of the velocity at \(r_{\rm{min}}\)?
    8. What is the tangential component of the velocity at \(r_{\rm{min}}\)?
    9. What is the value of the effective potential at \(r_{\rm{min}}\)?
    10. For what values of the initial total energy are there bound orbits?
    11. Using your results above, write a short essay describing this type of scattering problem, at a level appropriate to share with another Paradigm student.