## Activity: Projectile with Linear Drag

Theoretical Mechanics (4 years)
Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.
What students learn
• Practice solving separable differential equations
• Identify terminal speed $|\vec{v}_{term}|=mg/b$
• Horizontal motion is bounded
• Practice checking dimensions, considering limiting cases, and resolving graphs and equations
• Media

A ping pong ball is thrown from the roof of Weniger Hall with an initial velocity that makes an angle $\theta$ up from horizontal. The ball experiences a drag force that is proportional to its velocity.

Use Newton's 2nd Law to solve for the position of the ping pong ball for any value of time.

Hints:

• Consider the horizontal and vertical components of the motion separately.
• While you're planning and working on your solution, be prepared to answer the following questions:
1. What are you doing right now?
2. How will the result help you?
3. How are you checking that the result is sensible?

## Facilitation

This problem is too long to do in one go. I like to

1. give the prompt
2. let the students work in groups until most groups have thought about finding acceleration components
3. bring everyone together and lead a whole class discussion about finding the acceleration components
4. then let the students return to groups find the velocity components (solve the differential equations)
5. When most groups have finished this part of the problem, discuss the solution as a whole class. If possible, have a group present how they solved for one of the velocity components.
6. Lead a discussion about making sense of the velocity components - dimensions, special cases, graphing the functions and matching it to a conceptual story.
7. Repeat for the position components.

If you want to skips parts, skip position and just ask for velocity.

## Student Conversations

• Components: Most students will already be familiar with needing to break the problem into horizontal and vertical components from Introductory Physics.
• Direction of the Drag Force: Most students will need encouragement to draw free-body diagrams for multiple points along the motion to think about how the direction of the drag force changes during the motion. They are used to having a single freebody diagram from introductory physics.
• Velocity on a Freebody Diagram: Watch out for drawing the velocity on a free-body diagram. Encourage students to draw the velocity in another color and off to the side to distinguish it.
• Techniques for Solving Differential Equations: Some students might recognize (from a differential equations course) that they can solve the with integration factors. This absolutely can be done, but encourage the students to solve by separation to practice.
• Separating the Vertical Component of Velocity: Many students have trouble separating the differential equation for the vertical components of the velocity. They want to keep separate the drag and gravity.

\begin{align*} \frac{dv_y}{dt} +\frac{b}{m}v_y = -g \end{align*}

• Definite Integrals: Some students will want to do indefinite integrals with integration constants. Encourage them to do definite integrals so that the constant are easier to determine (you don't have to go back and plug in the initial conditions).
• group Box Sliding Down Frictionless Wedge

group Small Group Activity

120 min.

##### Box Sliding Down Frictionless Wedge
Theoretical Mechanics (4 years)

Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance $d$ down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.
• group Changing Spin Bases with a Completeness Relation

group Small Group Activity

10 min.

##### Changing Spin Bases with a Completeness Relation
Quantum Fundamentals 2023 (3 years)

Completeness Relations

Students work in small groups to use completeness relations to change the basis of quantum states.
• assignment_ind Normalization of the Gaussian for Wavefunctions

assignment_ind Small White Board Question

5 min.

##### Normalization of the Gaussian for Wavefunctions
Periodic Systems 2022

Fourier Transforms and Wave Packets

Students find a wavefunction that corresponds to a Gaussian probability density.
• assignment Phase 2

assignment Homework

##### Phase 2
quantum mechanics relative phase overall phase measurement probability Quantum Fundamentals 2023 (3 years) Consider the three quantum states: $\left\vert \psi_1\right\rangle = \frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle$ $\left\vert \psi_2\right\rangle = \frac{4}{5}\left\vert +\right\rangle- i\frac{3}{5} \left\vert -\right\rangle$ $\left\vert \psi_3\right\rangle = -\frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle$
1. For each of the $\left|{\psi_i}\right\rangle$ above, calculate the probabilities of spin component measurements along the $x$, $y$, and $z$-axes.
2. Look For a Pattern (and Generalize): Use your results from $(a)$ to comment on the importance of the overall phase and of the relative phases of the quantum state vector.
• assignment Pressure and entropy of a degenerate Fermi gas

assignment Homework

##### Pressure and entropy of a degenerate Fermi gas
Fermi gas Pressure Entropy Thermal and Statistical Physics 2020
1. Show that a Fermi electron gas in the ground state exerts a pressure \begin{align} p = \frac{\left(3\pi^2\right)^{\frac23}}{5} \frac{\hbar^2}{m}\left(\frac{N}{V}\right)^{\frac53} \end{align} In a uniform decrease of the volume of a cube every orbital has its energy raised: The energy of each orbital is proportional to $\frac1{L^2}$ or to $\frac1{V^{\frac23}}$.

2. Find an expression for the entropy of a Fermi electron gas in the region $kT\ll \varepsilon_F$. Notice that $S\rightarrow 0$ as $T\rightarrow 0$.

• assignment Potential energy of gas in gravitational field

assignment Homework

##### Potential energy of gas in gravitational field
Potential energy Heat capacity Thermal and Statistical Physics 2020 Consider a column of atoms each of mass $M$ at temperature $T$ in a uniform gravitational field $g$. Find the thermal average potential energy per atom. The thermal average kinetic energy is independent of height. Find the total heat capacity per atom. The total heat capacity is the sum of contributions from the kinetic energy and from the potential energy. Take the zero of the gravitational energy at the bottom $h=0$ of the column. Integrate from $h=0$ to $h=\infty$. You may assume the gas is ideal.
• assignment_ind Dot Product Review

assignment_ind Small White Board Question

10 min.

##### Dot Product Review
Static Fields 2023 (7 years)

This small whiteboard question (SWBQ) serves as a quick review of the dot product. It is also an opportunity to help students see the advantages of knowing many different representations of and facts about a physical concept.
• assignment Total Charge

assignment Homework

##### Total Charge
charge density curvilinear coordinates

Integration Sequence

Static Fields 2023 (6 years)

For each case below, find the total charge.

1. A positively charged (dielectric) spherical shell of inner radius $a$ and outer radius $b$ with a spherically symmetric internal charge density $$\rho(\vec{r})=3\alpha\, e^{(kr)^3}$$
2. A positively charged (dielectric) cylindrical shell of inner radius $a$ and outer radius $b$ with a cylindrically symmetric internal charge density $$\rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks}$$

• group $|\pm\rangle$ Forms an Orthonormal Basis

group Small Group Activity

30 min.

##### $|\pm\rangle$ Forms an Orthonormal Basis
Quantum Fundamentals 2023 (3 years)

Completeness Relations

Student explore the properties of an orthonormal basis using the Cartesian and $S_z$ bases as examples.
• assignment Contours

assignment Homework

##### Contours

Shown below is a contour plot of a scalar field, $\mu(x,y)$. Assume that $x$ and $y$ are measured in meters and that $\mu$ is measured in kilograms. Four points are indicated on the plot.
1. Determine $\frac{\partial\mu}{\partial x}$ and $\frac{\partial\mu}{\partial y}$ at each of the four points.
2. On a printout of the figure, draw a qualitatively accurate vector at each point corresponding to the gradient of $\mu(x,y)$ using your answers to part a above. How did you choose a scale for your vectors? Describe how the direction of the gradient vector is related to the contours on the plot and what property of the contour map is related to the magnitude of the gradient vector.
3. Evaluate the gradient of $h(x,y)=(x+1)^2\left(\frac{x}{2}-\frac{y}{3}\right)^3$ at the point $(x,y)=(3,-2)$.