# The Wagon Wheel Effect

Take a look at this helicopter hovering:

This is a much cooler version of an idea I discussed in a series of conference presentations I gave back in 1997 based around my interpretation of a super-cool effect I had first noticed as a kid when I watched old Westerns. I still think it is cool when I see images of stage coaches and wagons racing across the prairie in one direction when their wheels appear to spin in exactly the wrong direction. To show just how geeky I am, I was sitting on the beach back in the summer of 1997 digging my toes into the sand of Atlantic Beach, NC thinking about cool physical phenomena and some random math problems when I grasped the reason for the “backwards wheels” and also realized that I could model that behavior on my TI graphing calculators.

As with all good problem solutions, my analysis also led me to see that there were multiple speeds at which this backwards effect (I now know it is called a stroboscopic effect) happens, as well as an infinite number of speeds when the wheels will look like they are slowly spinning forward and between each of these effects, moments when the wheels appear to be motionless even though they are clearly spinning (as in the helicopter clip above). As a kid, I had been enamored with the backwards wheels, but after my fun on the beach, I began to look for (and see!) the whole range of effects on virtually every car commercial and movie with a vehicle in a chase scene.

This next video clip isn’t anywhere near as cool as the helicopter above, but it combines all of the wheel effects I first explored in the late 90s.

After my 2-day conference with Dan Meyer last week, I’m tempted to return to the scene of this new problem and add new hooks (perhaps the helicopter video above) to create new models explaining this stroboscopic effect. A future (hopefully soon) post will offer further insights. Until then, I’d love to know your thoughts and comments.

### 2 responses to “The Wagon Wheel Effect”

1. So my question here is which blade is spinning faster? I’m pretty puzzled by how one would go about finding the answer. If we assume the camera is filming at 30 fps, we can see the main blade must be spinning at $\frac{30n}{5}$ multiple of that, where n is an integer. But I’m not sure what do to take things further. It would have been awesome if they had filmed the same helicopter with a different frame rate, and then you’d probably be able to compare the two.

2. I had forgotten about this until reading the post. Do you still have the graphing calculator program I wrote for this? I vaguely remember something about it being published in a math journal?