vector – Rear Propulsion Steering Spline, or what is this called?

What I am looking for is essentially to know what this is, or what I can google to learn how to do it… unless of course the solution is simple.enter image description here

You have an object at the start position facing a direction, and a desired position and direction.

The object has:

  • max speed
  • rotation rate
  • acceleration
  • deceleration

I am essentially trying to simulate the object having a “thruster” force acting on it.

enter image description here

I would like a smooth path from VecStart to VecEnd, where the object accelerates from stationary, and then decelerates as it gets to its desired position and rotation.

I am at a loss for what this is called, and where I should begin to research.