Physics 303 Test 2

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Questions (5 pts. apiece) Answer in complete, well-written sentences WITHIN the spaces provided.

1. Consider a system for separating different atoms based on the kinetic energy of scattered protons. The energy of protons scattered from carbon and oxygen atoms $K_1$ is shown below. If the resolution of the detector is $\rm 20~eV$, then what is the best angular range for identifying the two atoms? Explain.

2. Is energy conserved in inelastic collisions? If not so, what happens to the lost energy? If so, what conditions must be satisfied?

3. What is the differential cross section?

4. Spacecraft launched from the Kennedy Space Center in Florida are always launched towards the east. Does the direction of the launch effect the kinetic energy needed to reach orbit (or are they just worried about hitting someone)? Explain.

Problems. Clearly show all reasoning for full credit. Use a separate sheet to show your work. See the next page for a table of equations and constants.

1.	25 pts.	A satellite is moving with a velocity $v$ in the Earth's upper atmosphere where the density is $\rho$. It has a cross-sectional area of $A$ perpendicular to the direction of motion. What is the drag force on the satellite in terms of $v$, $\rho$, and $A$? Assume the air molecules are moving slowly compared with $v$ and their collisions with the satellite are completely inelastic.
2.	25 pts.	Two atoms in a diatomic molecule with masses $m_1$ and $m_2$ interact through a potential energy $$V(r) = {a \over r^4} - {b \over r^3}$$ where $r$ is the separation of the atoms and $a$ and $b$ are positive constants. Assume circular orbits of angular momentum $l$ for all the questions below. (a) If the atoms move in circular orbits, then what is the equilibrium separation of the two atoms in terms of the constants above? (b) What is the maximum equilibrium separation the atoms can have before breaking up? Your answer should be in terms of $a$, $b$, and $\mu$. (b) What is the maximum angular momentum the molecule can have without breaking up? Your answer should be in terms of $a$, $b$, and $\mu$.

See next page.

3.

30 pts.

One of the early pictures of the atom was the Thomson model. The atom was viewed as a spherical cloud of uniformly distributed positive charge and mass with a radius of $\rm 1.0~\AA$. The electrons were fixed in place at points throughout the volume of this `atom'. Consider an $\alpha$ particle (a $\rm ^4He$ nucleus with $Z=2$) of energy $\rm 3.6~MeV$ that collides with a magnesium atom ($Z=12$) at rest. We will investigate the deflection of this $\alpha$ particle by the magnesium atom as predicted by the Thomson model. (a) For the moment ignore the effect of the electrons in the atom. Starting from the expressions listed in the table, what is the maximum deflection of the $\alpha$ particle created by the sphere of positive charge if the $\alpha$ particle does not penetrate the interior of the sphere? This is an overestimate of the deflection of the $\alpha$ particle before it penetrates the nucleus. (b) If the $\alpha$ particle does penetrate the sphere the effect of the positive charge is reduced and collisions with single electrons become important. Single electrons can deflect the $\alpha$ particles up to $0.008^\circ$ (but no more). What is the largest possible deflection that could be created by the twelve electrons in the magnesium atom? This is an overestimate of the deflection of the $\alpha$ particle after it penetrates the nucleus. (c) What is an estimate of the largest angle one could expect to see scattered $\alpha$ particles if the Thomson model was accurate? Is this result consistent with observations? Explain.

Some useful constants and relationships.

=15pt

${1 \over r} = {\mu \alpha \over l^2} \left (1 + \epsilon \cos (\theta - \theta_0) \right )$	$\epsilon = \sqrt{ 1 + {2 E l^2 \over \mu \alpha^2} }$	$\sin \left ( {\theta_s \over 2} \right ) = { 1 \over \epsilon}$
${ d\sigma \over d \Omega} = \left ( {\alpha \over 4 E_{cm}} \right )^2 { 1 \over \sin^4 \left ( {\theta_s \over 2} \right ) }$	$\alpha = -ke^2 Z_1 Z_2$	$\rm ke^2 = {\hbar c \over 137} = 1.44 ~ MeV-fm$
$\rm 1.0~ \AA = 10^{-10} ~m$	$\rm 1~fm = 10^{-15}~m$	$\rm 1~eV = 1.6\times 10^{-19}~J$
$e = \left \vert {\vec v_{2f} - \vec v_{1f} \over \vec v_{2i} - \vec v_{1i}} \right \vert$	$V_{eff}(r) = {l^2 \over 2 \mu r^2} + V(r)$