Teaching approximation techniques in basic courses

[andresB]

Not sure how universal it is, but my experience through half of my undergrad education gave me the impression that 90% physics was about exactly solvable problems. Off the top of my head, the only approximation we ever did in introductory courses was the binomial expansion to get the electric field of a dipole. Only at quantum mechanics 2 I encountered approximation methods being taught seriously.

As a teacher, I would like to teach approximation techniques almost/specially from the very beginning in the courses of introductory physics. I would like to give at least some examples here and there where mathematical approximations are required or at least greatly reduces the work needed to solve the problem at hand. But such examples are not that common in the standard textbooks.

So, I'm here for advices and to hear other people experiences. How do you handle approximating results when teaching introductory course of physics?

 

[Office_Shredder]

Is this a calculus based physics course or just an algebra one? Are the students going to understand how you use derivatives to compute tangent lines?

It's probably worth pointing out even if it's calculus based, they might struggle enough with the material that the tangent lines confuse them.

 

[andresB]

Is this a calculus based physics course or just an algebra one? Are the students going to understand how you use derivatives to compute tangent lines?

It's probably worth pointing out even if it's calculus based, they might struggle enough with the material that the tangent lines confuse them.

Uhm, around here the first semester of introductory physics is more algebra based but with the assumption that the student is taking calculus at the same time, and some student have already completed differential calculus. Think about Halliday and Sear-Zemansky books.

 

[Office_Shredder]

I would be pretty nervous about that situation. If you try to explain something like $\sin(x)\approx x$ you're going to have a pretty wide range of understanding. It might be worth it to let them solve pendulum problems by hand, but if you throw a lot of stuff like that at them a lot of them are doing to be spending half their study time trying to memorize these formulas.

 

[mpresic3]

I often taught the small angle approximation when I taught freshman physics in the first semester when we hit the peendulum. I remember a lab we did where we used the Newton Raphson method the first semester one year but we did not have that lab the second year. Multiple expansion involved Taylor series approximation (in several variables) the beginning of the junior year. We expanded the square wave in terms of Fourier Coefficients, sophomore year but this might be a bit unusual. Sometimes undergraduate physcs gets into normal mode problems that involve expressing a quadratic form with a small angle approximaton.

I have heard engineers criticize that physics just treats ideal cases. However, ideal cases form the basis for approximate cases. Usually in QM for example, you need the exact solution to the unperturbed problem (ideal case) to get the energy shift due to a perturbation.

BTW, I think no other dicipline gets as much mileage out of Taylor series, and first order approximation as physics does.

 

[atyy]

BTW, I think no other dicipline gets as much mileage out of Taylor series, and first order approximation as physics does.

Second order too, since the harmonic oscillator is studied over and over again :)

 

[vela]

I have heard engineers criticize that physics just treats ideal cases.

And the reason physicists look at ideal cases is because they're trying to discover the underlying principles, which can be difficult to see if one includes various complications in the analysis.

I'll also note that intro physics uses approximations right from the start. We approximate the motion of a baseball as free fall, but we all know there are forces other than gravity on the ball. Frictionless surfaces don't exist, but we often approximate the frictional force to be 0. These approximation all serve to simplify the problems; a series expansion is just another way to simplify a problem.

 

[andresB]

It's behind a paywall, but this seems like an interesting example
https://aapt.scitation.org/doi/10.1119/1.5097757

 

[Will Flannery]

"Not sure how universal it is, but my experience through half of my undergrad education gave me the impression that 90% physics was about exactly solvable problems."

Your impression was the result of an illusion. In reality very few problems in physics have exact solutions - even in mathematics not everything is exact. So, for example, when you use your calculator to compute a sine function, the value returned is an approximation.

When it comes to physical processes, the laws governing processes are written as differential equations, and most differential equations do not have closed form solutions. Take for example Newton's model for a falling object, you can see the analytic solution here ... https://en.wikipedia.org/wiki/Free_fall ... in the inverse square law section, the analytic solution is an infinite series ...

To see how I would introduce approximation into introductory physics, see ... www.berkeleyscience.com/TheComingRevA.pdf