## Student handout: A coarse-grained model for transportation

Contemporary Challenges 2022 (4 years)
A short lecture introducing the idea that most of the energy loss when driving is going into the kinetic energy of the air.

Let's start by visualizing the energy flow associated with driving a gasoline-powered car. We will use a box and arrow diagram, where boxes represent where energy can accumulate, and arrows show energy flow.

The energy clearly starts in the form of gasoline in the tank. Where does it go? Visualize the energy as an indestructable, incompressible liquid. “Energy is conserved”

The heat can look like

• Hot exhaust gas
• The radiator (its job is to dissipate heat)
• Friction heating in the drive train

The work contribute to

• Rubber tires heated by deformation
• Wind, which ultimately ends up as heating the atmosphere

The most important factors for a coarse-grain model of highway driving:

1. The 75:25 split between “heat” and “work”
2. The trail of wind behind a car
What might we have missed? Where else might energy have gone? We ignored the kinetic energy of the car, and the energy dissipated as heat in the brakes. On the interstate this is appropriate, but for city driving the dominant “work” may be in accelerating the car to 30 mph, and with that energy then converted into heat by the brakes.

• face Basics of heat engines

face Lecture

10 min.

##### Basics of heat engines
Contemporary Challenges 2022 (4 years) This brief lecture covers the basics of heat engines.
• assignment Power from the Ocean

assignment Homework

##### Power from the Ocean
heat engine efficiency Energy and Entropy 2021 (2 years)

It has been proposed to use the thermal gradient of the ocean to drive a heat engine. Suppose that at a certain location the water temperature is $22^\circ$C at the ocean surface and $4^{o}$C at the ocean floor.

1. What is the maximum possible efficiency of an engine operating between these two temperatures?

2. If the engine is to produce 1 GW of electrical power, what minimum volume of water must be processed every second? Note that the specific heat capacity of water $c_p = 4.2$ Jg$^{-1}$K$^{-1}$ and the density of water is 1 g cm$^{-3}$, and both are roughly constant over this temperature range.

• group Expectation Values for a Particle on a Ring

group Small Group Activity

30 min.

##### Expectation Values for a Particle on a Ring
Central Forces 2023 (3 years)

Quantum Ring Sequence

Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.
• group Energy radiated from one oscillator

group Small Group Activity

30 min.

##### Energy radiated from one oscillator
Contemporary Challenges 2022 (4 years)

This lecture is one step in motivating the form of the Planck distribution.
• group Black space capsule

group Small Group Activity

30 min.

##### Black space capsule
Contemporary Challenges 2022 (3 years)

In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.
• group Earthquake waves

group Small Group Activity

30 min.

##### Earthquake waves
Contemporary Challenges 2022 (4 years)

In this activity students use the known speed of earthquake waves to estimate the Young's modulus of the Earth's crust.
• face Wavelength of peak intensity

face Lecture

5 min.

##### Wavelength of peak intensity
Contemporary Challenges 2022 (3 years)

This very short lecture introduces Wein's displacement law.
• group Thermal radiation at twice the temperature

group Small Group Activity

10 min.

##### Thermal radiation at twice the temperature
Contemporary Challenges 2022 (4 years)

This small group activity has students reasoning about how the Planck distribution shifts when the temperature is doubled. This leads to a qualitative argument for the Stefan-Boltzmann law.
• assignment Vapor pressure equation

assignment Homework

##### Vapor pressure equation
phase transformation Clausius-Clapeyron Thermal and Statistical Physics 2020 Consider a phase transformation between either solid or liquid and gas. Assume that the volume of the gas is way bigger than that of the liquid or solid, such that $\Delta V \approx V_g$. Furthermore, assume that the ideal gas law applies to the gas phase. Note: this problem is solved in the textbook, in the section on the Clausius-Clapeyron equation.
1. Solve for $\frac{dp}{dT}$ in terms of the pressure of the vapor and the latent heat $L$ and the temperature.

2. Assume further that the latent heat is roughly independent of temperature. Integrate to find the vapor pressure itself as a function of temperature (and of course, the latent heat).

• face Equipartition theorem

face Lecture

30 min.

##### Equipartition theorem
Contemporary Challenges 2022 (4 years)

This lecture introduces the equipartition theorem.

Learning Outcomes