Here are a series of exercises, using math at levels ranging from algebra and geometry through calculus, needing no particular background in the sciences. A few of these need just some bare-bones ideas from science. Follow any link you wish. Enjoy!
How far is it worth traveling for cheaper gasoline? Some reasonable assumptions about vehicle fuel economy and a decent amount of algebra give us an estimate (it’s not very far!)
How fast can rockets using chemical reactions get going in free space? Calculus leads us to a speed that’s logarithmic in terms of mass used up…a fast rocket has to have a very small payload and frame, or else use “tricks” like slingshotting around planets (not covered here)
Is √2 “normal,” with the digits 0 through 9 appearing equally frequently? How about e, the base of the natural logarithms? Both are irrational numbers, but the only simple way to test is via a tour de force, running through the digits. Needed: a text file giving either number to many decimal digits, and a program to search for them. Here’s a Fortran 90 program (easy to adapt to any other language), a version compiled for Windows (change the run.txt extension to exe to run it; exe’s can’t be sent readily over the Internet, as being potentially dangerous executables), a text file with e to about 17,000 decimal digits, and a PDF of a spreadsheet from the analysis of several hundred digits of √2.
Fun estimating the height and size of clouds while riding a bike. This is mostly on ways to observe the clouds and their shadows, some geometry, and some mental math. It makes a bike ride pass with more interest than one expects.
The falling skydiver: how fast is he or she going at any time? There is a fairly simple expression for the drag on a body falling in air (“body” as a generic in the physics sense, not meaning “dead body!”). One can insert it into a simple differential equation that is “straightforward” to solve by separation of variables; along the way, I prove some useful relations. I also show how far off is the prediction of a simpler form often given for its simpler math.
Complex numbers, the exponential, and geometry. The exponential, ex, comes up often, and its first appearance in calculus can be a bit hard to comprehend. There are several ways to define ex, each adding some utility. I have a bit more extensive write-up on the exponential, nicely typeset rather than handwritten.The complex exponential is one of the first real uses of complex numbers (I rail against the New Mexico math standards that throw in complex numbers to say, “I’m so cute as a high-school mathematician,” when complex numbers really begin to be useful in much advanced contexts, such as electrical engineering). Forgive my handwriting (which I sometimes have trouble reading); the part I wrote up about complex numbers is in my execrable printing; it takes a long time to set type, either with MathType or LaTeX.
What’s the pattern in the runs of heads in a coin toss? Generate some long runs and check the formulas of probability theory. This is pretty simple but amusing. The text is a Fortran 90 program and its output.
A study in predatory lending – payday loans and their phenomenal interest rates. treating interest as compounded instantaneously (close to what the loan sharks do), I use the payment amounts and number of payments to solve for the total payment, which involves an effective interest rate. This occurs in a nonlinear equation, which I solve iteratively, introducing some students to Newton’s method.
A Fortran program to calculate the solar angles (zenith and azimuth) for any day and time. Set your latitude, the offset of civil time from local solar noon, and the Julian day, and get the angles for a range of hours in the day. If you wish not to compile a Fortran program nor to recode in, say, Python, I’ve also included an executable for Windows (change run.txt to exe to get it to run in a terminal). There’s sample output, too, for Las Cruces on the summer solstice. The program uses all the spherical trigonometry of the Earth’s rotation and its orbit (it includes the Equation of Time for the variation in angular speed of the orbit, which is elliptical, not circular).
The decay chain of U-238: how fast do those daughter products build up? The decay of each element is a simple exponential, coupled to the preceding one’s decay as input. So, there are n coupled differential equations, all first order, all linear, and the inhomogeneous solution is readily calculated analytically from the homogeneous solution. I just redid this post (30 November 2018) . The math now uses no assumptions about relative half-lives of elements. I’ve added much interpretation of the results. The decay chain sketch is now tidy and all on one page. I include an Excel program to calculate the amounts and activities of U-238 and the next three nuclides in the chain. The full write-up is here.
Probability: how many Volvo trucks do you expect to see when n trucks have passed? I explain a minor obsession of mine, counting trucks while riding my bike along a highway, and use it to illustrate some laws of probability and interesting calculations from them. Here you’ll find an explanatory text as well as an Excel program for calculating probabilities.
Fall of a ball in Earth’s gravity: simple formulas and an experimental test (I’ll gather the files shortly)