## Scattering

• assignment Graphs Involving the Distance Formula

assignment Homework

##### Graphs Involving the Distance Formula
Static Fields 2023 (6 years)

Learn more about the geometry of $\vert \vec{r}-\vec{r'}\vert$ in two dimensions.

1. Make sketches of the following functions, by hand, on the same axes: \begin{align} y &= \sin x\\ y &= \sin(2+x) \end{align} Briefly describe the role that the number 2 plays in the shape of the second graph
2. Make a sketch of the graph $$\vert \vec{r} - \vec{a} \vert = 2$$

for each of the following values of $\vec a$: \begin{align} \vec a &= \vec 0\\ \vec a &= 2 \hat x- 3 \hat y\\ \vec a &= \text{points due east and is 2 units long} \end{align}

3. Derive a more familiar equation equivalent to $$\vert \vec r - \vec a \vert = 2$$ for arbitrary $\vec a$, by expanding $\vec r$ and $\vec a$ in rectangular coordinates. Simplify as much as possible. (Ok, ok, I know this is a terribly worded question. What do I mean by “more familiar"? What do I mean by “simplify as much as possible"? Why am I making you read my mind? Try it anyway. Real life is not full of carefully worded problems. Bonus points to anyone who can figure out a better way of wording the question that doesn't give the point away.)
4. Write a brief description of the geometric meaning of the equation $$\vert \vec r - \vec a \vert = 2$$

• assignment Surface temperature of the Earth

assignment Homework

##### Surface temperature of the Earth
Temperature Radiation Thermal and Statistical Physics 2020 Calculate the temperature of the surface of the Earth on the assumption that as a black body in thermal equilibrium it reradiates as much thermal radiation as it receives from the Sun. Assume also that the surface of the Earth is a constant temperature over the day-night cycle. Use the sun's surface temperature $T_{\odot}=5800\text{K}$; and the sun's radius $R_{\odot}=7\times 10^{10}\text{cm}$; and the Earth-Sun distance of $1.5\times 10^{13}\text{cm}$.
• assignment Directional Derivative

assignment Homework

##### Directional Derivative

Static Fields 2023 (6 years)

You are on a hike. The altitude nearby is described by the function $f(x, y)= k x^{2}y$, where $k=20 \mathrm{\frac{m}{km^3}}$ is a constant, $x$ and $y$ are east and north coordinates, respectively, with units of kilometers. You're standing at the spot $(3~\mathrm{km},2~\mathrm{km})$ and there is a cottage located at $(1~\mathrm{km}, 2~\mathrm{km})$. You drop your water bottle and the water spills out.

1. Plot the function $f(x, y)$ and also its level curves in your favorite plotting software. Include images of these graphs. Special note: If you use a computer program written by someone else, you must reference that appropriately.
2. In which direction in space does the water flow?
3. At the spot you're standing, what is the slope of the ground in the direction of the cottage?
4. Does your result to part (c) make sense from the graph?

• group Scalar Surface and Volume Elements

group Small Group Activity

30 min.

##### Scalar Surface and Volume Elements
Static Fields 2023 (7 years)

Integration Sequence

Students use known algebraic expressions for length elements $d\ell$ to determine all simple scalar area $dA$ and volume elements $d\tau$ in cylindrical and spherical coordinates.

This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements.

• assignment Basic Calculus: Practice Exercises

assignment Homework

##### Basic Calculus: Practice Exercises
Static Fields 2023 (4 years) Determine the following derivatives and evaluate the following integrals, all by hand. You should also learn how to check these answers on Wolfram Alpha.
1. $\frac{d}{du}\left(u^2\sin u\right)$
2. $\frac{d}{dz}\left(\ln(z^2+1)\right)$
3. $\displaystyle\int v\cos(v^2)\,dv$
4. $\displaystyle\int v\cos v\,dv$
• assignment Entropy of mixing

assignment Homework

##### Entropy of mixing
Entropy Equilibrium Sackur-Tetrode Thermal and Statistical Physics 2020

Suppose that a system of $N$ atoms of type $A$ is placed in diffusive contact with a system of $N$ atoms of type $B$ at the same temperature and volume.

1. Show that after diffusive equilibrium is reached the total entropy is increased by $2Nk\ln 2$. The entropy increase $2Nk\ln 2$ is known as the entropy of mixing.

2. If the atoms are identical ($A=B$), show that there is no increase in entropy when diffusive contact is established. The difference has been called the Gibbs paradox.

3. Since the Helmholtz free energy is lower for the mixed $AB$ than for the separated $A$ and $B$, it should be possible to extract work from the mixing process. Construct a process that could extract work as the two gasses are mixed at fixed temperature. You will probably need to use walls that are permeable to one gas but not the other.

Note

This course has not yet covered work, but it was covered in Energy and Entropy, so you may need to stretch your memory to finish part (c).

• assignment Hockey

assignment Homework

##### Hockey
Central Forces 2023 (3 years)

Consider the frictionless motion of a hockey puck of mass $m$ on a perfectly circular bowl-shaped ice rink with radius $a$. The central region of the bowl ($r < 0.8a$) is perfectly flat and the sides of the ice bowl smoothly rise to a height $h$ at $r = a$.

1. Draw a sketch of the potential energy for this system. Set the zero of potential energy at the top of the sides of the bowl.
2. Situation 1: the puck is initially moving radially outward from the exact center of the rink. What minimum velocity does the puck need to escape the rink?
3. Situation 2: a stationary puck, at a distance $\frac{a}{2}$ from the center of the rink, is hit in such a way that it's initial velocity $\vec v_0$ is perpendicular to its position vector as measured from the center of the rink. What is the total energy of the puck immediately after it is struck?
4. In situation 2, what is the angular momentum of the puck immediately after it is struck?
5. Draw a sketch of the effective potential for situation 2.
6. In situation 2, for what minimum value of $\vec v_0$ does the puck just escape the rink?

• group Vector Surface and Volume Elements

group Small Group Activity

30 min.

##### Vector Surface and Volume Elements
Static Fields 2023 (4 years)

Integration Sequence

Students use known algebraic expressions for vector line elements $d\vec{r}$ to determine all simple vector area $d\vec{A}$ and volume elements $d\tau$ in cylindrical and spherical coordinates.

This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.

• group Work By An Electric Field (Contour Map)

group Small Group Activity

30 min.

##### Work By An Electric Field (Contour Map)

Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.
• assignment Free Expansion

assignment Homework

##### Free Expansion
Energy and Entropy 2021 (2 years)

The internal energy is of any ideal gas can be written as \begin{align} U &= U(T,N) \end{align} meaning that the internal energy depends only on the number of particles and the temperature, but not the volume.*

The ideal gas law \begin{align} pV &= Nk_BT \end{align} defines the relationship between $p$, $V$ and $T$. You may take the number of molecules $N$ to be constant. Consider the free adiabatic expansion of an ideal gas to twice its volume. “Free expansion” means that no work is done, but also that the process is also neither quasistatic nor reversible.
1. What is the change in entropy of the gas? How do you know this?

2. What is the change in temperature of the gas?

• 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.)

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.