How to integrate velocity squared to solve action integral?

[nabeel17]

Consider a forcefree mass point in one-dimensional space.
(a) Calculate the action S for the actual path of the mass point in the time interval
[0, T] and for the boundary conditions x(0) = 0 and x(T) = d.

I said the Lagrangian was just equal to L=1/2mv^2. I'm not sure if my reasoning for this is correct (I may have a conceptual error) but since there is no force acting on it, the potential is 0 (or a constant but it can be set to 0?)

so the action integral is S=∫1/2mv^2dt where the limits are 0-T

I'm not sure how to integrate v^2 with respect to t. Even if my approach is wrong, I would still like to know how that integral is done

 

[nabeel17]

oops, velocity would be constant...so that solves that. However, if it wasn't constant, how would I go about solving that integral?

 

[haruspex]

You cannot solve it without some information about how velocity varies with time.

 

[elfmotat]

You need to either know v(t) or v(x). If you know the second then you can use the fact that v²dt=vdx.

 

[nabeel17]

ahh, ok. Thank you