Determing approximate bicycle speed for distancemeter
I have built many years ago a device nearly identical to what you describe. I did not have the additional requirement of a minimum speed of 10 kph, however. As noted in the comments, that is likely best resolved with GPS as you are detecting motion, not absolute speed figures.

I used "arduino gps library" as the search terms and found many options. One of them, TinyGPS++ appears to have what you require and much more: 


  TinyGPS++ is a new Arduino library for parsing NMEA data streams
  provided by GPS modules.
  
  Like its predecessor, TinyGPS, this library provides compact and
  easy-to-use methods for extracting position, date, time, altitude,
  speed, and course from consumer GPS devices. 
  
  However, TinyGPS++’s programmer interface is considerably simpler to
  use than TinyGPS, and the new library can extract arbitrary data from
  any of the myriad NMEA sentences out there, even proprietary ones.


I would definitely appreciate the "considerably simpler" aspect, as well as the "extract arbitrary data" portion, which seems specific to your project.

There may be other libraries that provide your objective, this was the first on the search returns.

The device I constructed used the common ultrasonic detector with a transmitter and receiver pair, coupled to a set of seven-segment LED display blocks. When the slightly-less-than-one-meter distance was presented, my device would trigger a pair of cameras. For your beep purpose, it would be straight-forward to redirect the camera trigger to a piezo speaker module or any other suitable audio device. I did not require GPS or other speed information.