Since the ship is generating the black hole at a fixed distance in front of itself, the black hole will not move on it's own. So, when it's gravity pulls the ship forward, it moves with the ship. Whichever way the ship is pointing is where the black hole will be. BTW, we're not talking about a black hole with stellar mass. It's a small singularity with just enough gravity to draw the ship forward. It's a 'tame' black hole.
No such animal as a "tame" black hole. The forces required to create them and the amount of mass that has to be compacted is enormous. Yes, scientists have been able to generate small ones, but they are microscopic and wink out of existence in a very short time only 30 femtoseconds, which is a really short period of time.
Here........................educate yourself.........................
Scientists created a 'black hole' using this ultra-powerful laser
Quit being a prick. This is a hypothetical question, based on the ability to create a tame black hole. What I want to know is could a black hole that is artificially generated by a space ship be used to move the ship?
Once a black hole is created it moves how it will, based on the interaction of the various gravity wells it is near. As has been pointed out, there is no such thing as a tame black hole.
You need to read what I posted again. The black hole is generated at a fixed distance in front of the ship. As the ship is drawn towards it, it remains the same distance from the ship. So, when the ship moves, the black hole moves with it.
How are you going to fix the distance? Anything that is past the event horizon is affected by the black hole and drawn towards it, but when it hits the event horizon, it gets sucked in. Any black hole capable of generating enough gravity to move a ship is too big to be controlled by it.
Besides, to "fix" the distance between the ship and the black hole would require a force much stronger than gravity itself. And, if it is at a fixed distance, why would the black hole only draw the ship to itself? What about everything else around it? You'd be sucking in comets, meteors and space debris, which would result in a very perilous situation for those on the ship.
Again................nope, the idea won't work.