Time for a new setup!

Rolling wheels across the floor, I've noticed that especially at the end of their runs, they sometimes seem to pick up a bit of speed from nowhere, rolling several meters on the newfound energy. This shows up in the position-time graphs as a bit of a wobble that has made some data difficult to fit well, and makes for big uncertainties on the accelerations. Yesterday, this showed up when I tried a different analysis on the data. Fitting windows of 5 data points to a quadratic, I determined the acceleration for one of the runs, and got the following:

This seems to be common to all of the data. The average is pretty reasonable, but there's a lot more going on here...

The accelerations are terrible to start mostly because the wheels don't take long to cross one tile, so that quantization error is a big deal. This is aggravated by the fact that getting the acceleration from a position vs. time graph involves taking the second derivative, and derivatives generally degrade data. As a quick aside, this is something that I've always rationalized using Fourier transforms. If we have a function written as:

\[f\left(x\right)=a_0+\sum_{n}\left(a_{n}\cos\left(nx\right)+b_{n}\sin\left(nx\right)\right)\]

then its derivative is given by:

\[f'\left(x\right)=\sum_{n}n\times\left(-a_{n}\sin\left(nx\right)+b_{n}\cos\left(nx\right)\right)\]

Note that the higher frequency components (with big $n$) are multiplied by bigger and bigger numbers. This is true of both the data (which is okay) and the noise (not so okay), so that the data gets noisier and noisier looking the more derivatives we take. At any rate, the problem in the early data might be subdued, if not really solved, by taking more accurate data for position and time.

The oscillations later on are particularly concerning, though. They occur over a distance scale of a few metres, which rules out an unbalanced wheel as a culprit. With a radius of about 5 cm, a poorly balanced wheel should give oscillations about once every 30 cm. It seems more likely that these are due to irregularities in the floor. To confirm this, I've laid out a common coordinate system along the floor (choosing a certain tile edge to be zero, going from there, and using that coordinate system for all videos) and looked for commonalities in the oscillations between experiments. It's hard to tell if they have a lot in common -- maybe yes, maybe no.

Clearly I'm going to need to measure this differently if I want to pick out a difference of 0.005 m/s² in acceleration. I'm thinking homebrew photogates over a one metre test section, and I've ordered a handful of laser pointers to see if I can rig something that'll work well in a hallway (otherwise, stray light is always a pain). More on that soon, I hope!