Contemporary Challenges: Spring-2026
Homework 3 (SOLUTION): Due 9 Friday
- Hot showers and standard deviation
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(a) Estimate the energy used during a typical 10-minute shower in the United States. The dominant variables in this problem are the temperature rise of the water and the flow rate of the showerhead. Develop a coarse-grained model that incorporates these two variables. For simplicity, you may assume: (i) The water heater is 100% efficient at converting electrical energy into heat, and (ii) the water heater is located close to the shower, so heat losses in the pipes can be neglected.
(b) Develop a reasonable estimate for the standard deviation of your answer to part a. That is, imagine measuring the energy used for a 10-minute shower in a thousand randomly selected U.S. households. How much would the energy use vary? To develop a reasonable estimate, spend a little time doing internet research (or in-person research) on the variability of shower flow rates, the variability of ground temperature, and the variability of shower water temperature (personal preference). Once you can justify reasonable numbers for the variability of these inputs, propogate the uncertainty through your coarse-grained model using the methods we discussed in class.
(c) Sense making: Compare the typical energy used for a 10-minute shower to the typical energy used to drive an elecric car at 70 mph for 10 minutes.
Shower
The flow rate of a shower can vary significantly. The average flow rate is about 2 gal/min, but you can easily find showers using 1.5 gal/min to 2.5 gal/min. The standard deviation is about 0.5 gal/min (this is estimate).
The average 10-minute shower uses 20 gallon of water, which has a mass \(m\) = 80 kg, the standard deviation in water mass is \(\sigma_m=\) 20 kg. Therefore \(\sigma_m/m=\) 25%)
The water must be heated from ground temperature (pipes are buried in the ground where the temperature is about 12 C) to comfortable shower temperature (body temperature is 38 C, water burns skin at 50 C). Most people like the temperature in the range 38 - 42 C. Let's say the average value of \(\Delta T\) = 28 C, and the standard deviation is about 2 C. Therefore, \(\sigma_{\Delta T}/\Delta T=\) 10%)
The heat required is then \(Q = m c \Delta T\), where \(c = 4200\) J/(K.kg) and \(m\) is the mass of the water. \begin{align} Q_\text{avg} &= (80 \text{ kg})(4.2\times10^3 \text{ J/(K.kg)})(28 \text{ K}) = 10 \text{ MJ} \end{align}
Using pythagorium addition of the percentage uncertainties I get a final uncertainty of 27%. My final answer for a 10-minute shower in then \begin{align} Q &= 10 \pm 3\text{ MJ} \end{align}
If I refined my coarse-grained model, I could include the energy required to pump this water up a hill. For a hill of 100 m, let's compare the change in gravitational potential energy to the heat energy: \begin{align} \frac{PE}{Q} = \frac{m g h}{m c \Delta T} = \frac{(10\text{ m/s})(100\text{ m})}{(4200\text{ J/(K kg)})(30\text{ K})} = \frac{10^3}{126\times10^3} = \frac{1}{126} \end{align} For a hill of 100 m, the gravitational potential energy cost is less than \(1\%\) of the heat energy.
Car
In the class notes we estimated 20,000 J/s. Driving the car for 10 minutes would use \begin{align} \text{Energy used } = (2 \times 10^4 \text{ J/s})(6 \times 10^2 \text{ s}) \approx 10 \text{ MJ} \end{align}
The rate that we use electrical energy to heat shower water is very similiar to the rate that we use electrical energy to drive a car at 70 mph.
- Gasoline Sun
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Electromagnetic radiation energy from the Sun arrives at the upper atmosphere of our planet at a rate of about 1350 J/(s \(\cdot\) m\(^2\)). Use this information, together with the average radius of the Earth's orbit, to show that the Sun radiates energy at a rate of about \(4 \times 10^{26}\) J/s.
We want to know the rate that energy radiates from the Sun. At a distance of \(1.5 \times 10^{11}\) m, the energy arrives at a rate of \(1350\) J/(s \(\cdot\) m\(^2\)). By conservation of energy, the rate that at which the energy leaves the sun, must also equal the rate at which it passes through the sphere at radius \(r\) from the sun. Using that the Area of a sphere is \(4\pi r_{\text{orbit}}^2\), we can find the energy per time.
\begin{align} \frac{\text{Energy}}{\text{time}} &= (1350 \text{ J/s.m\(^2\)} )(4 \pi (1.5 \times 10^{11} \text{ m})^2) \\ &= 4 \times 10^{26} \text{ J/s} \end{align}
We know from radiometric dating of rocks on Earth (and the Moon and Mars) that our solar system is about 4 billion years old. Let's make a naïve hypothesis (like scientists did in the early 1900s) that the Sun is powered by burning hydrocarbons. What mass of gasoline would be needed to power the Sun at a rate of \(4 \times 10^{26}\) J/s for 4 billion years? Compare to the actual mass of the Sun.
Note: The energy density of hydrocarbon fuels, including gasoline, natural gas, dry logs of wood, chocolate, croissants, gummy bears, etc. etc. is \(\approx\) 40 MJ/kg.
Total energy emitted as light during the 4 billion year lifetime: \begin{align} E_{\text{light}} &= (4 \times 10^{26} \text{ J/s})(3\times 10^7 \text{ s/y})(4\times 10^9 \text{ y}) \\ &= 50\times10^{42} \text{ J} = 50\times10^{36} \text{ MJ} \end{align} How much gasoline would it take to produce this energy? \begin{align} m_{\text{gasoline}} = \frac{50\times10^{36}\text{ MJ}}{40 \text{ MJ/kg}} = 1.25\times10^{36} \text{ kg} \end{align} We can compare this to the mass of the sun, \(M_{\text{sun}} = 2\times 10^{30}\) kg. The sun would have to be almost 1 million times more massive if it was running on hydrocarbon fuels.
- Practice Nuclear Physics
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1. Rutherford scattering
Based on Q13M.2 from Unit Q 3rd Edition
Estimate the minimum kinetic energy that an alpha particle must have to get stuck to a gold nucleus? Give your answer in both joules and electron volts.(Note: The atomic number of gold is \(Z = 79\), the mass number of gold is \(A = 197\). An alpha particle consists of 2 protons and 2 neutrons. This question can be solved using knowledge of the Coulomb potential for a system of two point charges).
An alpha particle can get inside a gold nucleus if its initial kinetic energy is large enough to overcome the electrostatic repulsion between the positively charged gold nucleus and the positively charged alpha particle. We need to calculate the electrostatic potential energy when the surface of an alpha particle is about to touch the surface of a gold nucleus. The nuclear radius of gold is estimated from \(r_{\text{Au}}=r_{0} A^{1 / 3}=7.27\ \mathrm{fm}\) where \(A=197\) is the mass number and \(r_0 = 1.2 \text{ fm}\). The nuclear radius of the alpha particle is estimate by \(r_{\text{alpha}}=r_{0} 4^{1 / 3}=1.9\ \mathrm{fm}\). Therefore, the kinetic energy that the alpha particle needs is \begin{align} KE_{\alpha}&=k \cdot \frac{2 e \cdot 79 e}{ r_{\text{Au}}+r_{\text{alpha}}}\\ &=4 \times 10^{-12} \text{ J}\\ &=25\text{ MeV} \end{align}
where \(k\) is the Coulomb's law constant \(9 \times 10^9\) J.m/C\(^2\).
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2. Fusion
- The figure shows the fusion reaction between tritium and deuterium. Estimate the electrostatic potential energy of a deuterium-tritium pair at separation distance where the strong nuclear force becomes significant (i.e. the nuclei "touch" and the strong force can stick them together). Give a numerical answer in joules.
- Fusion is happening in a hot ionized gas of deuterium and tritium (also known as a D-T plasma). The average kinetic energy of the individual deuterium and tritium ions is about 10 times smaller than your answer to part a. (Fusion can be sustained by a small fraction of nuclei that have higher-than-average kinetic energy). Estimate the temperature of the plasma using the relationship:
Average kinetic energy of gas atom \(=(3/2)k_BT\),where \(k_B = 1.38 \times 10^{-23}\) J/K is called the Boltzmann constant. Give a numerical answer in kelvin.
- Compare the mass of \(^3\)He (a nucleus with two protons and one neutron) to the combined mass of two isolated protons and one isolated neutron. You are welcome to reference emperical data, but the main task of this question is to estimate the mass difference using the coarse-grained model we discussed in class. In this coarse-grained model, the biggest contributions to binding energy are the electrostatic interaction and the strong interaction.
- Estimate the mass difference and give your answer in units of kilograms.
- Is the mass of the composite particle larger or smaller than the mass of the individual parts?
- Using the approximation method for finding nuclear radius: \(r_1=1.51\) fm and \(r_2=1.72\) fm. \begin{align} U_{electric}&=k \cdot \frac{e^2}{ [3.23 \times 10^{-15} \text{ m}]}\\ &= 7.1 \times 10^{-14} \text{ J} \end{align} where \begin{align} k=9\times 10^9 \text{ J.m/C}^2\\ e=1.6\times 10^{-19} \text{ C}\\ \end{align}
From equipartition theorem (which we will study later), we expect each particle in a gas to have kinetic energy \begin{align} KE \approx (3/2)k_BT\\ \end{align} We are looking for a temperature where an average deuterium or tritium ion has \(KE \approx 7.1 \times 10^{-15} \text{ J}\). Therefore, \begin{align} T &= (2/3) \frac{[7.1 \times 10^{-15} \text{ J}]}{3k_B}\\ &= \frac{2 \cdot [7.1 \times 10^{-15} \text{ J}]}{3\cdot [1.38 \times 10^{-23} \text{ J/K}]}\\ &= 3.4\times 10^{8} \text{ K}\\ &= 340 \text{ million K} \end{align}
- Solution hasn't been writen yet.
- Three ideas for the term project
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Read the description of the term project on the class website at “Introduction to term project”. Identify three (3) subjects that you find interesting/intriguing (for example, solar energy, exoplanets, ...). Within each subject, pose a question that might have an interesting quantitative answer: “Since it requires energy to make a solar panel, how long does it take to recoup that energy?”, “How far away could we see an Earth-like planet orbiting a Sun-like star?” ... You should turn in 3 different subjects and 3 different quantitative questions (quantitative means “quantities that can be calculated and/or measured”)
Let your mind wander as broadly as possible. Subjects and questions are not restricted to the topics taught in PH315. During this exploratory stage, be bold and daring; you are not committing yourself to solve all 3 questions. To spark your imagination, there is a list of ideas on the class website. The instructor will read your ideas and give you feedback. Whenever possible, the feedback will point you towards a coarse-grained model that is helpful for answering your question. Use the feedback to help decide which question you will develop further (or whether you need to go back to the drawing board).
Here are some general comments which the instructor might reference when giving individual feedback.
Cost estimates
General advice: If an accountant could solve your question, you haven't included enough physics.
There are important/interesting relationships between physics and what stuff costs. For example, if you understand the rocket equation, and you know the price of rocket fuel, you can set a minimum cost for launching payloads into space. The cost of launching payloads into space is a great example of how we can connect physics to the real world. However, please keep the physics in your question/solution and avoid focusing exclusively on tallying up dollar amounts.Topics in alphabetical order
Airplane flight: The textbook "Sustainable Energy Without the Hot Air" gives some excellent models about airplane energy consumption (see the technical appendices that explain the mathematical models at the end of the book). These models allow you to explore the effect of changing the weight and velocity of the aircraft, as well as the effect of changing the air density or even the force of gravity (flight on Mars?).
Cars and delivery trucks: The energy used for stop-and-go driving can be described by a coarse grain model. See MacKay textbook for details. Differences in operating cost between gas-powered cars and electric cars are interesting. Note that city driving (coarse-grained model) is strongly affected by vehicle weight, but highway driving is nearly independent of vehicle weight (except for rolling resistance).
Desalination: There is a simple physical model to calculate the energy required to desalinate water. The work done is PdV, just like gas processes. The osmotic pressure, P, can be written in the same form as the ideal gas law, PV = nRT, where n is the number of moles of solute molecules. Despite being different physical systems (one is a gas, the other is salt dissolved in water), they share the same mathematical description because the entropy of the solute molecules in liquid is identical to the entropy of gas molecules in an ideal gas. The entropy explanation for osmotic pressure is a bit abstract. Therefore, it's interesting to think about a physical model of what happens on the microscopic level. How would the salt water interact with a semipermeable membrane? Here is a link to such a model: “OSMOSIS: A MACROSCOPIC PHENOMENON, A MICROSCOPIC VIEW” https://journals.physiology.org/doi/full/10.1152/advan.00015.2002
Direct-Air-Capture of CO2: You could make a physical model of direct-air capture of CO2. You have to blow air across the surface of water. CO2 can dissolve in water. Chemicals in the water can bind to the dissolved CO2. The tricky part of the calculation is finding the work done driving laminar air flow through a constricted space. Learn about the viscosity of air and the physics of laminar flow. Useful comparisons include: How much CO2 does a tree capture? How much CO2 does seaweed capture?
Drones, Helicopters, Honeybees and Humming Birds: There is a coarse-grain model to figure out how much energy it takes for a flying vehicle to hover in mid-air. The spinning blades (or honeybee wings) generate a downdraft. This downward moving air has momentum. The rate of momentum generation (kg.m/s per second) is equal to the force that counteracts gravity. Once you do the momentum calculation, you can then figure out energy used: \(\frac{1}{2}mv^2\), where \(m\) is the mass of air used in the downdraft, and \(v\) is the velocity of the downdraft.
Earth-like planets in the Milky Way galaxy: The Drake equation is a popular way to estimate whether life exists outside our solar system (there is a good Wikipedia page about the Drake equation). You can use a simple climate model (from Unit T) to estimate the range of orbital distances that yield a good planet temperature near a sun-like star. This adds some interesting physics to the calculation of probabilities that enter the Drake equation. Also, there is always new data to consider about earth-like planets. For example: https://www.universetoday.com/146583/astronomers-estimate-there-are-6-billion-earth-like-planets-in-the-milky-way/.
Energy used by a Data Center: It's interesting to think about the energy a computer uses to process information. You need a course-grain model of the physical process used inside computers. At the heart of a computer are MOSFETs (a type of transistor). A MOSFET is a parallel plate capacitor (the channel is one plate, the gate is the other plate). There are about 1 billion transistors in a processor. Some fraction of them switch every cycle (1 GHz cycling rate?). When a transistor switches, it dumps the energy stored in the capacitance \(\frac{1}{2}CV^2\). https://www.intel.com/pressroom/kits/core2duo/pdf/epi-trends-final2.pdf .
Fluid flow in pipes: Hagen-Poiseuille equation allows you to develop simple models for the work done moving liquid along a pipe.
Food production: For each type of crop, you can look up an efficiency fraction = (food energy)/(solar energy). There are maps of solar energy as function of location and month. For data on Oregon, I like this website: http://solardat.uoregon.edu/NorthwestSolarResourceMaps.html.
Food consumption: Humans eat food and utilize that chemical energy at a rate of about 80 J/s. In the United States, humans use their other sources of energy at an average rate of about 8000 J/s. It's a big difference! There are multiple course-grained methods to estimate the rate 80 J/s. This can be an excellent topic for the term project. Course-grain methods include (i) radiative heat transfer to the environment, (ii) food consumption, (iii) the volume of air we breathe. (The oxygen we consume is used to "burn" carbohydrates and power our bodies...)
General Relativity: The wikipedia page on "gravitational redshift" discusses some historical controversy that could make good material for a question. It is possible to estimate the Schwarzschild radius by simple methods (see class notes) Video by minute physics: "the unreasonable efficiency of black holes": https://www.youtube.com/watch?v=t-O-Qdh7VvQ. Heat transfer (e.g. temperature of a spaceship) In many cases you can ignore convective heat transfer and just focus on radiative heat transfer. Radiative heat transfer models give lots of insight and are easier to implement mathematically than convective heat transfer. See UnitT6.4 for some guidance about the Stefan-Boltzmann law, and example questions at the end of the chapter such as T6M.4. Radiative heat transfer explains how a spaceship or satellite maintains a steady temperature. Radiative heat transfer gives a good estimate for energy consumed by a human.
Heat transfer (e.g. heating a meteorite from the outside): The heat equation is a PDE with so many useful applications. https://en.wikipedia.org/wiki/Heat_equation The university education of an engineer can involve years studying all the different approaches to solving this equation. In the context of this class, we need a quick first-order (approximate) analysis. A general approach that I recommend: imagine your system as slabs of finite thickness and uniform temperature. For example, a sphere heated from the center might be modelled like the layers of an onion. Energy flows from hotter slabs to colder slabs. Before the system reaches steady state, the transient behavior might be tricky to model. You'd have to take finite time steps and use a program like excel, matlab or python to help you. When the system reaches steady state, the math should get easier.
Hydroelectric power: Questions about hydroelectric can be “rich context” if you apply to a specific situation. For example, you might start with basic info like the annual rainfall and catchment area of the river, and compare to the population and per capita energy consumption. For making a simple model (including things like rainfall catchments), there is guidance in the "Sustainable Energy Without the Hot Air".
Hyperloop: The hyperloop concept is a nice system to analyze. There are good resources available on the internet. There are multiple questions you can ask. For example, how the energy usage scales with air pressure inside the tube, what are the concerns about making the system safe, how much energy is used to maintain the vacuum.
Ionizing radiation (alpha, beta and gamma radiation): Chpt 15 of Unit Q discusses some physical models of biological effects of radiation. I've also written a question along these lines, which I'd be happy to share as a starting point. Maximum dose needed for medical imaging is another interesting angle to think about.
Information technology: About 5% of electricity in the US is being used for data centers, computers, internet, streaming data etc. Interesting coarse grain models include energy used per bit of information transmitted down a fiberoptic cable, and the energy used to process one bit of information using a modern integrated circuit. https://www.intel.com/pressroom/kits/core2duo/pdf/epi-trends-final2.pdf .
Laser weapons: https://en.wikipedia.org/wiki/Laser_weapon Wikipedia as a detailed overview, including a chart describing over 50 military projects/experiments. The Locust system is the most recent in 2026.
Methane as a greenhouse gas: To calculate the potency of methane, you'll need to consider the current concentration of methane in the atmosphere, and calculate the optical pathlength for IR light that is resonant with methane vibrations. If you feel comfortable with such calculations, this would be a great question. The MODTRAN atmosphere simulator will also be helpful. I have class-notes from week 9 that will be helpful (happy to share them with you).
Oceans: Energy from temperature differences in Ocean: There is a small OTEC plant operating in Hawaii (OTEC = ocean thermal energy conversion plant. As a physicist you can do a zeroth-order analysis of this specific system and then start asking questions like: would the system work in Oregon?; can the system be scaled up?; what is the energy cost of making the cold-water intake deeper in the ocean?
The heat engine in an OTEC plant uses ammonia as the working gas and takes heat from a hot reservoir (warm water) and dumps heat into a cold reservoir (cold water). Please don't get bogged down in trying to learn how engineers build a closed-loop ammonia system and how expanding ammonia gas is used to do work on a turbine (the turbine replaces the piston we discussed in class). It takes years of engineering school to design the inner workings of an industrial heat engine. If you are worried about fine tuning your calculation, you can always change this “engineering factor” from 70% of Carnot efficiency to 80% of Carnot efficiency or vice versa.
Neutron stars: Some of the concepts that are covered in the nuclear physics section of the course are relevant for analyzing neutron stars.
Pulsars: The mechanism of pulsar luminosity hasn't been figured out yet. However, there are still fun calculations related to pulsars. We know that a pulsar is a spinning object and a pulse from a pulsar happens every 5 milliseconds or so. What is the maximum radius of an object that is spins this fast yet is still held together by gravity? Can we put limits on the mass density of a pulsar?
Quantum Computing: It's difficult to explore quantum computing without learning a lot more quantum mechanics and the associated mathematical language/techniques. However, there are accessible entry points. Physicists like Nicolas Gisin have developed wonderful and insightful back-of-the-envelop calculations related to Bell's theorem. I recommend reading Jeffery Bub's most recent book “Totally Random” (or his earlier book “Bananaworld”). Both e-book versions are available from OSU library. Bub describes a pair of entangled quantum coins that produce correlated coin toss results. You could write a homework question related to Bub's proof that the quantum coins cannot be rigged to mimic the observed entanglement correlation. “Totally Random” also explains a number of applications for quantum entanglement which are accessible to PH315 students. Please note that entanglement is the true resource required for quantum computing to outperform classical computing. Therefore, a solid understanding of entanglement is a critical first step before understanding quantum computing.
Quantum Mechanics applied to Astrophysics: It's possible to estimate the Chandrasekhar mass by fairly simple methods (see wikipedia article). Chandrasekhar mass is the largest mass for a stable white dwarf, it is governed by the pressure it takes to squeeze an electron into a small space (compressing the deBroglie wavelength to very small size). Similar physics goes into calculating the largest possible mass of a neutron star (squeezing the deBroglie wavelength of neutrons). People argue about the exact upper limit to the neutron star's mass because you have to consider how general relativity interacts with quantum mechanics. This gets complicated!
Sky Hooks and Space Elevators: The term project webpage has links to YouTube video about sky hooks (space tethers). A word of caution, however. In past years, students have struggled with space tethers because it's hard to simplify the calculations. The equations get messy very quickly. It's hard to design a simple thought experiment to illustrate how the proposed systems will work. I haven't found good resources to guide students beyond the superficial YouTube video.
Solar power in Oregon: There are some great maps of oregon solar energy at this website: http://solardat.uoregon.edu/NorthwestSolarResourceMaps.html. If you write an Oregon solar energy question, consider referring to these maps in your question. Space travel: laser-powered sail An important challenge is the divergence of the laser beam. You should look at Figure Q3.9 "A graph of intensity vs. sin(theta) for waves going through a single slit..." The accompanying text is "Q3.5 Diffraction Revisited".
Space travel: rockets: Take a look at "the rocket equation" (Wikipedia). If you are mostly concerned with escaping Earth's gravity, you'll also need to read about escape velocity on Wikipedia. Using today's technology, the launch mass of a rocket is mostly rocket fuel... the payload is only a small fraction. Also, note that the amount of energy required for space travel depends on the propulsion system (mass of the propellent, velocity of the propellent).
Space travel: changing orbit: If you want to move a heavy object into a different orbit, you'll need a detailed step-by-step plan. It's not satisfactory to simply say “I'll magically change the gravitational potential energy and the kinetic energy”. For a specific step-by-step plan, consider using the Hohmann transfer orbit: https://en.wikipedia.org/wiki/Hohmann_transfer_orbit or you can compare Hohmann to something else. If you want to calculate the energy required, please remember that the amount of energy required to move something by rocket propulsion depends on the propulsion system.
Superconductors at high pressure: This is an exciting topic. Here is an article about recent advances in the research field: https://www.nature.com/articles/d41586-020-02895-0. The prediction by Neil Ashcroft is very relevant. He noted that hydrogen (pressed into solid form) is the best choice for getting superconductivity at room temperature (because hydrogen is light weight and it has a half-filled electron shell). You could look up Ashcroft's prediction/equations and try to write your question about it.
Stars: The rate of energy emission (electromagnetic radiation) from a star can be calculated from the Stefan-Boltzmann law if you know the star's surface temperature and radius. See UnitT6.4 for some guidance about the Stefan-Boltzmann law.
Transmission line energy loss: Estimating the energy loss over transmission lines is a good question. Some transmission is done at 200 kV, some at 50 kV, how thick are the cables? how long are the cables? What fraction of the energy transmitted is wasted?
Wind: The internet has useful maps showing the average wind speed around the United States. https://windexchange.energy.gov/maps-data/325
Wireless power transmission (far-field techniques): Read about wireless power transfer on Wikipedia, especially the subsection about far-field techniques. You should be able to calculate the size of the transmitting antenna and the receiving antenna for a given situation. The textbook Unit Q will help you understand the main physics concepts. See Figure Q3.9 A graph of intensity versus sin(theta) for waves going through a slit.