Okay, this is something that is tripping me out. It must be wrong but I can't figure out what.

Say you have a rocketship in outer space with a particular thrust. Assume for simplicity that thrust and vehicle mass are constant throughout the experiment. Because of this constant thrust, the rocketship asymptotically approaches a certain maximum speed s.

After getting arbitrarily close to maximum speed, the rocketship rotates exactly 90 degrees so that its thrust is now perpendicular to its velocity. Because nothing is slowing the ship down along its original heading, the ship's overall speed asymptotically approaches s*sqrt(2) because its speed along the new heading approaches s just like the speed along the old heading did.

Perhaps you see the problem now. If the ship applies thrust in successively perpendicular directions, its speed will increase to s*sqrt(3), s*2, s*sqrt(5), etc. So regardless of thrust magnitude, the ship can attain any speed, and it does so with skewed thrust rather than by going straight!

So can someone explain, with at least some rigor, what is wrong with my reasoning?

MT,

The velocity is produced due to the thrust:
then how could their directions differ???

Could you plz review the Question,
or
enlighten me in this school of Physics???

I don't know about rigor, but it seems to me that the acceleration is proportional to the force and inversely proportional to the mass. When you change directions, even though you continue to have the same velocity forward, you can't continue or renew acceleration. That particular mass has already accelerated to its max. In order to add 2 accelerations vectorially, wouldn't they have to take place at the same time? At least that's my guess.

I think I have the answer. There is no asymptotic limit to the ship's speed (in the Newtonian sense) because thrust is constant. Acceleration is not dependent on the ship's speed because the velocity at which reaction mass escapes the engines is relative to the velocity of the ship.

Why didn't I think of that yesterday? It seems so clear now.

Why should there be a maximum speed? (Unless you mean lightspeed, of course.) Terminal velocity only applies when there's a force such as air friction acting against acceleration.

Yes, JoMS. I must have posted *just* before you.

I'm not sure why you've defined the thrust as assymptotically approaching a constat speed, and I don't think that, in space at least, there should be any reason for this. But i do see one problem with your reasoning beyond this. you state in your question "Because nothing is slowing the ship down along its original heading"
this is key. You can rotate 90 degrees and provide thrust in such a way that you are not changing the velocity in the original direction. In 3 dimensional space, you can then find another direction to accelerate in that won't affect either of the previous velocities. after that, you're stuck. any direction you chose from now on can be defined as some combination of one of the directions you have already accelerated in.