*simple* example of mechanics problem with no closed-form solution?

[bcrowell]

I'm looking for a good example for a freshman mechanics class to demonstrate how one can integrate the equations of motion numerically when there is no closed-form solution. The problem below is the best I've been able to come up with yet, but I'm not totally happy with it, and I'm wondering if anyone can suggest a better one. Here are my criteria:

(1) It should be of the form where essentially the acceleration (or force) is given as a function of time from t1 to t2, and we want to integrate to find either v or x. It's also OK if the force is velocity-dependent.
(2) It should be physically simple, and it should not depend on any knowledge of vectors, two-dimensional motion, energy, or Newton's law of gravity. Preferably it would not even assume any previous knowledge of topics like static and kinetic friction.
(3) There should be no closed-form solution in terms of elementary functions like exponentials, trig functions, and logs.
(4) It should be a problem that is physically interesting, natural, and well-motivated -- not just something where I write down some expression that happens to be impossible to integrate in closed form.
(5) I would also like to be able to come up with a second variation on the same problem where we want to find the time at which x or v reaches some value of interest, and I would like this variation to be physically interesting and well motivated.

My Antarctic rescue example below fails criterion 2, mainly because it involves two-dimensional motion.

I'd be grateful for any suggestions!

BTW, if you want to figure out whether a function is not integrable in closed form, an easy way to do it is to go to integrals.com and type it in.

A really natural example is the motion of a falling object subject to a force from air friction that is proportional to v^2. However, this example fails criterion #3, since there is a closed-form solution (although it is not easy to find).

Thanks in advance!

-Ben

==============

The pilot of a small airplane crashes in Antarctica in a storm and activates her personal locator beacon.
A rescue helicopter is unable to land due to whiteout conditions near the ground, but will drop
a supply package to her known location from the minimum height yo=30 m to which it can safely descend.
The package is blown sideways by the wind, so the helicopter, hovering at height yo,
needs to release it from a point displaced by a horizontal distance x.
Based on experiments by Von K\'arm\'an (1881-1963),one typically
expects that wind speed is proportional to y^{1/7}, and one also usually finds that air friction
depends on the square of the speed, so that the horizontal force on the package is given
by ky^{2/7}, where k is a constant. Let k=10 in SI units, and let m=1 kg. Find x.

The vertical motion is known in closed form,
<br /> y = y_\zu{o}-\frac{1}{2}gt^2 \qquad .<br />
We then have a horizontal acceleration given by
<br /> a = \frac{F}{m} = \frac{k}{m}\left(y_\zu{o}-\frac{1}{2}gt^2\right)^{2/7} \qquad .<br />
This acceleration will be applied from t=0 to t=\sqrt{2y_\zu{o}/g}.

 

[IsometricPion]

The dynamics of a pendulum are non-linear and can be described in terms of a single parameter. Though, in order to motivate the equation I guess you might need to use vectors and do a force diagram.

Maybe a non-linear version or time-varying version of the spring equation motivated by non-Newtonian fluids (perhaps a 1-D version of the http://en.wikipedia.org/wiki/Maxwell_material" models).

 

[bcrowell]

Hi, IsometricPion,

Thanks for the suggestions! However, they both seem more physically complicated (less easily understandable to a freshman physics student) than the one I've already got.

-Ben

 

[AlephZero]

A comment on your own example: you are using air resistance to create the horizontal force, but you ignored it in the vertical direction. "Illogical, Captain?"

A simple system with surprisingly complicated dynamics is a ball bouncing vertically above a plate that is doing simple harmonic motion at a given frequency. You can ignore air resistance. The collisions are simple to model since the motion of the plate is prescribed. You don't need to bring in momentum and energy, just the idea of a coefficient of restitution if you want to take energy out of the system and get to a "steady state" situation (which may not be at the same frequency as the plate is oscillating).

 

[A. Neumaier]

I'm looking for a good example for a freshman mechanics class to demonstrate how one can integrate the equations of motion numerically when there is no closed-form solution.

Take the quartic anharmonic oscillator. You can't find anything simpler.

 

[homology]

that's simpler than a pendulum?

 

[A. Neumaier]

that's simpler than a pendulum?

Yes. You don't need to know about trigonometric functions, and you can do perturbation theory without needing to know about Taylor expansions. And since it is well-known that Hooke's law is valid only for small deviations from equilibrium, it is easy to motivate the cubic term in the force as a correction to Hooke's law, together with a symmetry assumption that eliminates a possible quadratic term.

Thus 12 year olds can understand it, if you give them an intuitive understanding of the meaning of the first two derivatives, how to compute them for powers, and how to add successive corrections to an approximation.

 

[bcrowell]

Thanks, all, for the ideas!

An anharmonic oscillator is indeed a physically simple example, but to really appreciate it, you need to already understand the harmonic oscillator, which my students wouldn't at this stage. In fact, I already assign that as a homework problem for numerical computation, but later in the course. Many of my students don't know about Taylor series at that point, so it takes a little work to explain to them why we would introduce the higher-order terms. In the homework assignment, I have them play around and see the effect of turning the anharmonicity on and off, see how it breaks the amplitude-independence of the frequency, and how you recover the s.h.m. behavior at low amplitudes.

Aleph Zero has a good point about the inconsistent treatment of the vertical and horizontal forces of air friction. The justification would be that I'm just assuming that the air friction force is relatively small, so the horizontal deflection is a small effect, and so is the effect on the vertical motion. Actually, that probably isn't even true for the numbers I made up for the example, but in any case, I want to get rid of that example.

The ball bouncing above the plate is a cool example, but it doesn't really suit my purposes, since the motion breaks down into pieces, with each piece being solvable in closed form.

I think my original example can be reworked in order to eliminate the issues with horizontal and vertical motion and the need for an approximation in order to neglect vertical friction. For example, you can get the same kinds of von Karman 1/7-power forces in other contexts, like water flowing through a pipe. However, I'm having trouble coming up with anything that isn't totally artificial. I would really like something that has some real-world interest.

The real challenge here is that these are community college students I'm teaching, they're only a few weeks into a freshman course, and this example is one they're supposed to read immediately after being exposed to Newton's second law for the first time.

 

[AlephZero]

Many of my students don't know about Taylor series at that point, so it takes a little work to explain to them why we would introduce the higher-order terms.

Get them to measure the force-displacement curve for an elastic band. Or if that doesn't grab their attention enough, use the elastic from a sufficiently non-politically-correct item of clothing...

 

[Dale]

There are a lot of problems that have an analytical solution in one set of coordinates, but not in another. One of those might be good since they would allow you to do a "reality check" with the exact solution.

 

[bcrowell]

Here's an example I think may work. We have a meteor entering the Earth's atmosphere vertically. Gravity is negligible compared to air friction, which we model as being proportional to v^2. Air pressure depends on altitude like e^-y. The equation of motion is something like a=v^2e^-y, which doesn't seem to me to be something that's likely to have a closed-form solution, although I could be wrong about that.

Get them to measure the force-displacement curve for an elastic band. Or if that doesn't grab their attention enough, use the elastic from a sufficiently non-politically-correct item of clothing...

Ah, I smell a sexual harassment lawsuit :-)