What sensor to inspect for object motion detection in an open but limited space?
Trouble with ultrasound is that it doesn't stay in the box - you still have reflections from background. So for objects that are more than a few tens of cm from the sensor you need to do signal processing to find the target. In which case you might as well pick a solution using visible light to make it easier to debug. Computer vision tracking of moving objects against static background is a solved problem; see OpenCV for a free implementation. It works best against neutral backgrounds, but doesn't require that. Two cameras gives you stereo imaging and three positions.

If you just want ball speed along an axis you can get off-the-shelf radar guns for this purpose. Building your own radar system is an extremely difficult prospect.

Edit: 50m/s should have been in the original spec, that's quite fast. Ultrasonic usually works by sending out "chirps" and timing the response. Speed of sound is ~340m/s. So it takes 35ms for sound to traverse the 12m box. During that time a 50m/s target has moved 1.75m. You need to wait until your first pulse has gone out and back before sending the next, so pulse interval is at least 70ms. So you can only get one measurement every 3.5m. 

You should also spec the latency: are you trying to interact with the moving object, or is it sufficient to work out where it was after is has passed? Do you care about the exact trajectory, or just speed of passing through the box (which can be found with doppler methods with ultrasound or radar).