Reddit mentions of Thunder Tumbler Radio Control 360 Degree Rally Car Red

I think you're on the right track. If I were to build it (and I'm no expert by any means), rather than using a knock sensor I would put a simple button switch on the front of the car to switch the motor - that way you ensure it makes contact with the wall rather than a "false knock" while it's traveling.

As far as going back to the point of origin, based on your original question of recording the distance, a stepper motor would give you precise control over revolutions and allow you to count with great accuracy how many revolutions it took to reach the wall. Given that there is a red circle for origin and second stopping point, I would use the color sensor to return to origin, ultimately saving money on not having to buy the more expensive stepper motors versus a simpler servo that you can just tell to "go until stopped".

As far as materials, you're correct with what you need. I don't know what the requirements for the project are, but your best bet might be to buy an inexpensive remote control car (I just took apart this specific RC truck to use as a rock tumbler. It has two 5v DC motors and the front wheels are "locked" so the car wouldn't turn on you. It takes 4 AA batteries, giving you 6v to power your arduino which is enough for this project)- with that you get a

• Body frame to house your components
  
• DC motor
  
• gear box
  
• wheels
  
• battery housing
  
  Remove the shell of the car, gut out the existing circuit board and toss on an arduino nano, wire the motor(s) to the arduino, wire and mount the color sensor to the bottom of the car (instructions), mount a switch to the front of the car, then start sketching.
  
  edit: formatting
  
  second edit: I was just thinking about this again - regardless of if you go with a knock sensor or switch, you'll want to power the motors with a single AA battery. They're made to spin pretty fast (not sure of the specific RPM's), so you really need to get them to slow down so the vehicle doesn't travel so fast that is "bounces" when it hits the wall. This would have the potential to knock it off course for the reverse travel.
  
  Speaking of reverse travel, I forgot to add the part about reversing the polarity to the motor. You'll need to switch the negative and positive lines to make the DC motor spin the opposite direction. The cheapest way that I can think of to accomplish this would be to add some transistors to the lines so you can switch (+) and (-). It'll take a couple of additional pins on the arduino, but you have them to spare.
  
  third edit: Or you could use a relay - I did this on a past project. Get two NO relays and have the normal conditions grounded with power running to the closed positions, then have the DC motor connected to the common pins of the relay. One side of the coil should be connected to ground with the other side connected to a pin on the arduino. When you set a pin to HIGH, it'll flip that relay to (+) and the car will move forward. Set that pin to LOW and the other pin to HIGH and the car will drive in reverse. Here's what I'm talking about (sorry it's so crude, I should be working right now :) ).
  
  fourth edit: I was talking to some people here at work and they asked why not just use an h-bridge for switching the motor direction. Again, not an expert, but most h-bridges switch 4.5v and higher which is more than you'd want to send to most RC motors unless they are low RPM motors to begin with. RC's are meant to go relatively fast, so the motors might be upwards of 6500 RPM with just a couple of reducing gears in the gear box.