How do I use ultrasound to track a moving object?
Yes you can use ultrasound to track a moving target, but I really doubt it could be done with an Arduino.

You need to be able to ping the object at least 3 times in the time it takes to move your beam width.  3 times because if you know where it is with the centred ping you need to calculate the direction of travel by pinging either side of the centre to see if it is there.

You then need to track it in three dimensions.  To do this you will need multiple sensors ideally at least 3 for 2D and 3 for 3D, so at least 6.  3 sensors in a plane will give you a reasonable degree of accuracy, but you beam width is key to this, the narrower the better.

Now you have to triangulate the position of the object, well actually work out the area of probability (for both planes) you never 'know' where it is really.

Oh and all 6 sensors need to have different frequencies to prevent interference, you then need to add in reflection processing to reduce false signals.

Its just much easier to buy a Pixy camera and download the software that they have for visually tracking your balls.