As someone who did a similar build in high school (though with a screen - no fancy VR smartphones back then): a cheap laser mouse actually works quite well as a speed measuring device, and requires no electronics hacking. I just calculated the speed of the cursor and translated that into game speed - and reversing comes for free :)
We also had a turning detector using a rod to connect the front-wheel to a 8€ USB joystick. It actually end up costing us ~20€, not counting the PC, bike and stand.
The benefit of taping a piece of paper to the wheel is that you could apply the same technique to something like a treadmill where you wouldn't want a magnet to be traveling along with the tread. The benefit of Arduino is lack of OS overhead/risk.
If you are going with a visual piece of tape, then you could replace the arduino with a simple mobile app that recognised the tape via camera and transmits speed via UDP to the app. I wonder if the camera on a typical phone could keep up ?
You could do the same with an inductive sensor wired to the headphone jack. This worked for Square to read magstripe data, which is significantly more complex of a signal to parse than simply counting the number of local maxima per second.
Inductive sensors are cheaper, simpler and more foolproof; cyclists have used them for decades to gather both wheel speed and cadence.
Slightly off topic:
You make an interesting point about Square.
In the context of Apple Pay, the move to take away the headphone jack suddenly makes a lot of sense (for Apple).
I don't know about that; the cheap Square reader is a piece of crap that is really just a "starter" product (they're not well-made and fall apart after even moderate use). Anyone who swipes more than 2 or 3 cards a day would be better served by one of Square's commercial readers -- all of which connect via Wi-Fi, Bluetooth or the Lightning port.
Also, Apple would certainly include a Lightning->Headphone adapter that would also be usable for card readers. They just use the line in from the microphone.
No photos, unfortunately, just our craptastic Powerpoint presentation :)
Now that I think about it, the school should have pictures. Unfortunately its site moves from platform to platform every few years, losing all content every time. But hey, Joomla is cutting edge, amirite?
https://drive.google.com/file/d/0B2WvkUwBYadXMTExNzNENzMyQjg... (In Portuguese. And wow, this is actually embarrassing. We only got a B- for that, and fully deserved. One of the other groups built a really cool electrostatic loudspeaker, and the other analyzed the traffic flow in our largest bridge with fluid dynamics simulations.)
We also had a turning detector using a rod to connect the front-wheel to a 8€ USB joystick. It actually end up costing us ~20€, not counting the PC, bike and stand.