Yeah, or a good statistics class. Real data is always messy. Although it might be a fun way just to generate some numbers to put on a graph for an early graphing lesson. I find that many students tend to do better with numbers for graphing if they have a concrete meaning attached to them.

]]>I’m thinking to get anything real out of it, maybe a pre-calc class? Unless all you want is the data collection.

]]>For our experiment, we pulled the car back 1 inch, 2 inches, 3 inches, etc. all the way up to 12 inches.

My son’s distances traveled (in inches): 1, 7, 12, 18, 25, 36, 48, 57, 60, 68, 85, 82.

My distances traveled: 1, 11, 14, 15, 26, 35, 44, 51, 68, 64, 75, 85.

One issue to note: the data is fairly noisy. For each of us, we even had one interval where the data wasn’t increasing. If you’re going to use this experiment, be prepared to deal with this in discussion.

The data looks convincingly close to linear on a log-log graph. So that was nice. Also, the measurement errors are such that small distances have very large relative errors. You may want to start, not at 1 inch, but at 3 or 4 inches.

]]>Oh! I see, you said between linear and quadratic. Duh. I get it now. That’s cool. I will mull this more.

]]>You can estimate C and p from data, but looking at the data using a log-log scale. This is because, if we take logarithms of the equation, we get (using the laws of logarithms)

log y = log C + p log x

That is, log y is a linear function of log x. So if we plot the data on a graph where the axes use log scales, the data should lie approximately on a line, where the slope is p, and the intercept is log C.

]]>That’s such a coincidence; I was rereading this post and wondering about Vroom Vroom. So you’re saying if we did it a bunch of times at varying distances we’d get a predictable function, but…exponential?

]]>First idea: when you pull the car back, you’re storing potential energy in a spring, and the energy stored in a spring is proportional to the square of the distance.

Second idea: when you release the car, the potential energy in the spring is converted to kinetic energy, with the conversion essentially complete at the starting line of the car. Kinetic energy is proportional to the square of velocity, so by the time the car reaches the original starting line, all the energy in the spring has been released, the velocity at that point will be proportional to the distance the car was pulled back.

Third idea: after the car passes the original starting line at a velocity proportional to distance pulled back, it will coast along, slowing because of friction, until it slows and stops. The only forces acting on it are frictional ones. Calculating how far it will go requires solving a differential equation, which will require deciding how the frictional term behaves. In general, we would expect the magnitude of the friction to be a function of the current velocity of the car: F(v).

If F(v) is a constant (i.e., it in fact doesn’t depend on velocity), you can solve the differential equation, and find that the distance traveled is proportional to the square of the initial velocity, and so proportional to the square of the distance pulled back, and the graph is a parabola.

If F(v) is proportional to v, on the other hand, you can still solve the differential equation, but you’ll now find that the distance traveled is proportional to the initial velocity, and so proportional to the distance pulled back, and the graph is a line.

A more realistic frictional term would probably be somewhere between these, and so the graph is likely to be approximately a power law (at least for reasonable pull-back distances), between linear and quadratic. Estimating the exponent would be doable from data; it’s a standard statistical regression problem. (The data would lie approximately on a line if you plotted it on a log-log scale, and the slope of the log-log graph would give you the exponent of the power law.)

It’d be a fun experiment to run, though, with plenty of mathematical and statistical content which could be appropriate for students at lots of levels if handled properly.

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