Fast-moving objects near our earth have different paths in the sky. They are different with stars far away.
For telescopes, when we do not know an object's orbit, catching up with its orbit and not losing it is a problem ? How to figure out a new object's orbit parameters quickly before we lose it? For an old object with an irregular orbit, can a telescope track it?
The typical speed in the asteroid belt may be smaller than 1"/min? But there may be faster objects. Then the paths are not so simple and regular.