Students from the University of Leicester have published a beautiful short research paper (pdf) on the physics of curving a soccer ball through the air.

It has been found that the amount a football bends depends linearly on the speed of the ball and the amount of spin.

— Sandhu et al., 2011: How to score a goal (pdf) in the University of Leicester’s Journal of Physics Special Topics

They derive the relationship from Bernoulli’s equation using some pretty straightforward algebra. The force (F) perpendicular to the ball’s motion that causes it to curl is:

and the distance the ball curls can be calculated from:

where:

F = force perpendicular to the direction the ball is kicked

D = perpendicular distance the ball moves to the direction it is kicked (the amount of curl)

R = radius of the ball

ρ = density of the air

ω = angular velocity of the ball

v = velocity of the ball (in the direction it is kicked)

m = mass of the ball

x = distance traveled in the direction the ball is kicked

The paper itself is an excellent example of what a short, student research paper should look like. And there are number of neat followup projects that advanced, high-school, physics/calculus students could take on, such as: considering the vertical dimension — how much time it take for the ball to rise and fall over the wall; creating a model (VPython) of the motion of the ball; and adding in the slowing of the ball due to air friction.

Energy cannot be either created or destroyed, just changed from one form to another. That is one of the fundamental insights into the way the universe works. In physics it’s referred to as the Law of Conservation of Energy, and is the basic starting point for solving a lot of physical problems. One great example is calculating the average temperature of the Earth, based on the balance between the amount of energy it receives from the Sun, versus the amount of energy it radiates into space.

The Temperature of Radiation

Anything with a temperature that’s not at absolute zero is giving off energy. You right now are radiating heat. Since temperature is a way of measuring the amount of energy in an object (it’s part of its internal energy), when you give off heat energy it lowers your body temperature. The equation that links the amount of radiation to the temperature is called the Stefan-Boltzman Law:

where:
E_{R} = energy radiated (W/m^{-2})
T = temperature (in Kelvin)
s = constant (5.67 x 10^{-8} W m^{-2} K^{-4})

Now if we know the surface area of the Earth (and assume the entire area is radiating energy), we can calculate how much energy is given off if we know the average global temperature (the radius of the Earth = 6371 km ). But the temperature is what we’re trying to find, so instead we’re going to have to figure out the amount of energy the Earth radiates. And for this, fortunately, we have the conservation of energy law.

Energy Balance for the Earth

Simply put, the amount of energy the Earth radiates has to be equal to the amount of energy gets from the Sun. If the Earth got more energy than it radiated the temperature would go up, if it got less the temperature would go down. Seen from space, the average temperature of the Earth from year to year stays about the same; global warming is actually a different issue.

So the energy radiated (E_{R}) must be equal to the energy absorbed (E_{A}) by the Earth.

Now we just have to figure out the amount of solar energy that’s absorbed.

Incoming Solar Radiation

The Sun delivers 1367 Watts of energy for every square meter it hits directly on the Earth (1367 W/m^{-2}). Not all of it is absorbed though, but since the energy in solar radiation can’t just disappear, we can account for it simply:

Some if the light energy just bounces off back into space. On average, the Earth reflects about 30% of the light. The term for the fraction reflected is albedo.

What’s not reflected is absorbed.

So now, if we know how many square meters of sunlight hit the Earth, we can calculate the total energy absorbed by the Earth.

The solar energy absorbed (incoming minus reflected) equals the outgoing radiation.

With this information, some algebra, a little geometry (area of a circle and surface area of a sphere) and the ability to convert units (km to m and celcius to kelvin), a student in high-school physics should be able to calculate the Earth’s average temperature. Students who grow up in non-metric societies might want to convert their final answer into Fahrenheit so they and their peers can get a better feel for the numbers.

What they should find is that their result is much lower than that actual average surface temperature of the globe of 15 deg. Celcius. That’s because of how the atmosphere traps heat near the surface because of the greenhouse effect. However, if you look at the average global temperature at the top of the atmosphere, it should be very close to your result.

They also should be able to point out a lot of the flaws in the model above, but these all (hopefully) come from the assumptions we make to simplify the problem to make it tractable. Simplifications are what scientists do. This energy balance model is very basic, but it’s the place to start. In fact, these basic principles are at the core of energy balance models of the Earth’s climate system (Budyko, 1969 is an early example). The evolution of today’s more complex models come from the systematic refinement of each of our simplifications.

Advanced Work

If students do all the algebra for this project first, and then plug in the numbers they should end up with an equation relating temperature to a number of things. This is essentially a model of the temperature of the Earth and what scientists would do with a model like this is change the parameters a bit to see what would happen in different scenarios.

Feedback

Global climate change might result in less snow in the polar latitudes, which would decrease the albedo of the earth by a few percent. How would that change the average global temperature?

Alternatively, there could be more snow due to increased evaporation from the oceans, which would mean an increase in albedo …

This would be a good chance to talk about systems and feedback since these two scenarios would result in different types of feedback, one positive and one negative (I’m not saying which is which).

Technology / Programming

Setting up an Excel spreadsheet with all the numbers in it would give practice with Excel, make it easier for the student to see the result of small changes, and even to graph changes. They could try varying albedo or the solar constant by 1% through 5% to see if changes are linear or not (though they should be able to tell this from the equation).

A small program could be written to simulate time. This is a steady-state model, but you could assume a certain percent change per year and see how that unfolds. This would probably be easier as an Excel spreadsheet, but the programming would be useful practice.

Of course this could also be the jumping off point for a lot of research into climate change, but that would be a much bigger project.

References

Yochanan Kushnir has a page/lecture that treats this type of zero-dimesional, energy balance model in a little more detail.

Another nice resource that provides neat graphs of real data that are easy for students to understand is Pollster.com. The graphs of survey results are constantly updated and, if you want to, you can go into how they were created (survey questions, averages etc.). They’re great for current event discussions and research projects.

In addition to the national polls, like the president’s job approval (see below), the site also has charts for state level races, like for governor, which are handy around election time.

Pollster.com aggregates polls, because, depending on how a question is phrased, each poll will have it’s own bias. However, since not all of the poll data is freely available to the public, the sites of the major polling organizations, like GALLUP, are also quite useful. The polling organizations tend to have a much wider variety of poll results available. Gallup in particular provides some very nice graphs.

Unemployment data for the Eastern US from GeoFRED.

GeoFred produces graphs like the one above showing economic statistics across the country. It can do it on a state by state basis, by county (as above), or by even more refined areas.

FRED plots graphs and even provides the data for economic statistics over time. The graph above shows GDP since they started collecting data in the 1940’s. It also has the times of recessions shaded in. The data in the graph can also be downloaded if you want to do your own analysis.

These seem to be handy little tools for teaching and for research projects, and are pretty easy to use.

The teachers’ training sessions are free though you have to get your own housing. They do provide breakfast and lunch. If you want graduate credit for them however, each three day session will cost around $306 and garner one credit.

Answers to the question 'Some people say that it is better for a country if different racial and ethnic groups maintain their distinct customs and traditions. Others say that it is better if these groups adapt and blend into the larger society. Which of these views comes closer to your own?' sorted by year in which the respondent was born.

UC Berkley’s Survey Documentation and Analysis (SDA) website has a lot of potential as a research tool for the more advanced middle-schooler. I greatly encourage students to do original research in their semester-long Independent Research Projects. They pose questions, collect and/or analyze data and slog through the challenges of dealing with open ended questions. Middle school is the appropriate time for this as they are working on their formal thinking skills. With the increasing availability of websites like the SDA, everyone can gain access to research grade datasets.

The SDA is powerful because it has a lot of data from survey questions dealing with a large number of survey issues, from race relations, to perceptions of the economy, to use of the web. But with that power is a certain degree of complexity. It took me a while this morning to decipher the web interface and I’m no where near plumbing all the nuances of the statistical analysis, but it’s not too hard to do some basic plots.

Right side of the SDA webpage.

On the left side of the window is a list of all the survey questions available. There are a lot but they each have the full question so it’s pretty easy to figure out what they mean. When you select one, such as the opposition to a family member bringing home a black/negro friend for dinner, it gives you the little code, “RACDIN” in this case that you enter as the Row on the right side of the window (see the above figure). Now I want to know how people’s answers to that question changed based on how old they are, so for the Column option I put in “AGE”. Of course what I actually put in is “AGE(c:10,1)” which tells the program to lump all the age data into 10 year sets, starting at age 1.

Answer to the question, 'How strongly would you object if a member of your family wanted to bring a (negro/black) friend home to dinner?' sorted by age of the respondent.

Students will certainly need help getting started, and I could add video instructions if anyone wants it.

Now comes the most interesting part, interpreting the graphs. I like the plot at the top of this post for this reason. It shows that the younger people are the more likely they are to think it’s better if racial and ethnic groups maintain their customs and traditions. Does this mean that younger people have more racist attitudes, trying to maintain separation, or does it mean that they are more accepting of different cultures?

In researching the benefits of sleep to assist a student interested in sleep deprivation, I came across the work of Matthew Walker from UC Berkley. His papers have some rather intriguing titles. One of them (Walker and van der Helm, 2009), called “Overnight Therapy? The Role of Sleep in Emotional Brain Processing” finds that lack of sleep seems to result in us retaining more negative memories and emotions, and reduces our ability to act rationally (Yoo et al., 2007):

[S]tudies indicate that prior sleep loss significantly impairs the ability for effective next-day learning of new experiences across numerous species. Furthermore, sleep loss appears to disrupt the learning of different affective categories to varying extents, potentially creating an imbalance in negative emotional memory dominance. – Walker and van der Helm, (2009)

Walker’s also synthesized the current research on sleep and finds sleep is important in other types of memory as well. During sleep we organize information in our brains, finding context for the data we absorbed while awake, which is why we remember things better after we sleep.

[S]leep serves a … role in memory processing that moves far beyond the consolidation and strengthening of individual memories and, instead, aims to intelligently assimilate and generalize these details offline. – Walker (2009)

Walker’s bibliography is a treasure trove of information about research on the affects of sleep on our emotional and intellectual well being. These articles will, however, probably need a lot of translation for the typical middle schooler. For that reason, using the press reports on this work would be a much better alternative for students.