Massless string pulled by a force

[russ_watters]

And if I say that I am fine with infinite acceleration, then what would the tension be?

This just results in a math fail. It's like asking if God can create a rock so big he can't lift it. It doesn't have an answer. Indeed, even stating that you apply a force to this string is impossible. So you're going to have to decide what you want to get out of this problem.

 

[Delta2]

In reality the string will have a small mass density ρ . Assuming that is moving with a finite acceleration a (because for example it is attached to other main bodies that do so) a length dx of the string will have a net force F=(ρdx)a=T(x+dx)-T(x)=dT as it follows from Newton's 2nd law. If the acceleration a is not too big such that the product ρa is small we can approximately say that the net force is zero from which it follows that dT=0 that is the tension along the string remain constant.

Finally i want to say something general that i find quite important when studying physics. It is quite often when studying physics that we do silent simplifying assumptions that simplify the calculations a lot and brighten only the key points of a phenomenon.If we don't make those assumption the study can become quite complex sometimes, so complex that we can't make any usefull conclusions.