Exercises

1. A particle subject to a repulsive force varying as $1/r^3$ is projected from infinity with a velocity that would carry it to a distance $a$ from the center of force, if it were directed toward the latter. Actually, it is projected along a line whose closest distance from the center of force would be $b$ if there were no repulsion. Prove that the particle's least distance from the center is $\sqrt{a^2+b^2}$, and that the angle between the two asymptotes of its path is $\pi\,b/\sqrt{a^2+b^2}$.
2. A particle subject to a repulsive force varying as $1/r^5$ is projected from infinity with a velocity $V$ that would carry it to a distance $a$ from the center of force, if it were directed toward the latter. Actually, it is projected along a line whose closest distance from the center of force would be $b$ if there were no repulsion. Show that the least velocity of the particle is
   
   $$V\,\frac{b^2}{a^2}\left[\left(\frac{a^4}{b^4}+\frac{1}{4}\right)^{1/2}-\frac{1}{2}\right]^{1/2}.$$
   
   

3. Using the notation of Section 6.2, show that the angular momentum of a two-body system takes the form
   
   $${\bf L} = M\,{\bf r}_{cm}\times \dot{\bf r}_{cm} + \mu\,{\bf r}\times \dot{\bf r},$$
   
   
   where $M=m_1+m_2$.
4. Consider the case of Rutherford scattering in the event that $m_1\gg m_2$. Demonstrate that the differential scattering cross-section in the laboratory frame is approximately
   
   $$\frac{d\sigma}{d\Omega'}\simeq \frac{q_1^{\,2}\,q_2^{\,2}}{4... ...rm max})^2}]^2\,\left[1-(\psi/\psi_{\rm max})^2\right]^{1/2}},$$
   
   
   where $\psi_{\rm max}=m_2/m_1$.
5. Show that the energy distribution of particles recoiling from an elastic collision is always directly proportional to the differential scattering cross-section in the center of mass frame.
6. It is found experimentally that in the elastic scattering of neutrons by protons ($m_n\simeq m_p$) at relatively low energies the energy distribution of the recoiling protons in the laboratory frame is constant up to a maximum energy, which is the energy of the incident neutrons. What is the angular distribution of the scattering in the center of mass frame?
7. The most energetic $\alpha$-particles available to Earnst Rutherford and his colleagues for the famous Rutherford scattering experiment were $7.7$MeV. For the scattering of 7.7MeV $\alpha$-particles from Uranium 238 nuclei (initially at rest) at a scattering angle in the laboratory frame of $90^\circ$, find the following (in the laboratory frame, unless otherwise specified):
   1. The recoil scattering angle of the Uranium nucleus.
   2. The scattering angles of the $\alpha$-particle and Uranium nucleus in the center of mass frame.
   3. The kinetic energies of the scattered $\alpha$-particle and Uranium nucleus (in MeV).
   4. The impact parameter, $b$.
   5. The distance of closest approach.
   6. The differential scattering cross-section at $90^\circ$.
8. Consider scattering by the repulsive potential $U=k/r^2$ (where $k>0$) viewed in the center of mass frame. Demonstrate that the differential scattering cross-section is
   
   $$\frac{d\sigma}{d\Omega } = \frac{k}{E\,\pi}\, \frac{1}{\sin\theta}\, \frac{(1-\theta/\pi)}{(\theta/\pi)^2\,(2-\theta/\pi)^2}.$$