assignment Homework
Central Forces 2023 (3 years)
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.)

What is the initial angular momentum of the system?

What is the initial total energy of the system?

What is the distance of closest approach \(r_{\rm{min}}\)
with the interaction?

Sketch the effective potential.

What is the angular momentum at \(r_{\rm{min}}\)?

What is the total energy of the system at \(r_{\rm{min}}\)?

What is the radial component of the velocity at \(r_{\rm{min}}\)?

What is the tangential component of the velocity at \(r_{\rm{min}}\)?

What is the value of the effective potential at \(r_{\rm{min}}\)?

For what values of the initial total energy are there bound orbits?

Using your results above, write a short essay describing this type
of scattering problem, at a level appropriate to share with another
Paradigm student.