Security cameras have a standard called "pixels per foot" (ppf). This means that at a certain distance, an object 1 foot across (about the size of a US license plate) will have a certain number of pixels. As the object moves further away, the number of pixels per foot decreases. As the object moves closer, the number of pixels per foot increases.
This is the correct way to measure whether you can read license plates or not, because it takes the camera lens field-of-view into account. This is particularly important when dealing with the GoPro since it has a maximum field-of-view of 170 degrees! This means its pixels are spread over a much larger area.
Conveniently, security cameras makers also have a standard for reading license plates of 40-45 pixels per foot.
Take a GoPro recording at 1920x1080 (aka 1080p) @ 170 degrees. Presumably you'd use this if mounted to the bike; the wider angle simply records more. Your resolution will drop to 45 ppf at 14.4 feet away. That is, beyond 14.4 feet away, it will get harder and harder to read license plates clearly.
A more thorough listing of a few different cameras:
- GoPro 1920x1080 @ 170 degrees; 14.4 ft
- GoPro 1920x1080 @ 127 degrees; 19.2 ft
- GoPro 1920x1080 @ 90 degrees; 27.2 ft
- GoPro 4k 3840x2160 @ 170 degrees; 28.8 ft
- GoPro 4k 3840x2160 @ 127 degrees; 38.5 ft
- GoPro 4k 3840x2160 @ 90 degrees; 54.3 ft
- Novatek NT96650 G1W 1920x1080 @ 120 degrees; 20.4 ft
- Mobius ActionCam 1920x1080 @ 116 degrees; 21.1 ft
The last two were selected because they were reviewed by Popular Mechanics as car dash cams, not because of their suitability for use on a bicycle.
More math! (oh noes!) A car travelling at 30 mph (relativistic) is travelling at 44 ft per second. That is, a car approaching you at 44 ft per second across a 30 ft distance (picked at random) will take 0.68 seconds to reach you. At 30 frames per second, that means you can record 20.5 frames. The more frames you get, the more likely that you get a clear one.
However, after laying all that math out, I'm not sure that 40 ppf is the minimum. Digital Image Forensics can recover an amazing amount of detail out of a blurry image, particularly because license plates are high contrast. Judge for yourself by searching for "digital forensics license plate"
I'm also not sure that 30 mph is a reasonable target to aim for; it seems a bit fast in most cases.
I recommend trying it out for yourself. Get a tape measure, borrow a camera, and record your license plate in 5 ft increments, and judge for yourself what seems like a effective distance is. Take angle into account; not everyone is going to be approaching the camera straight on; they could be approaching you from a cross street, so you might be viewing it at 45 degrees or so.