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My question is essentially this: is the following procedure correct?

a=-kx and we want v.

Multiply both sides by v to get:

[tex] \frac {d^2x} {dt^2} * \frac {dx} {dt} = -kx *\frac {dx} {dt} [/tex]

Now, by dt, and some rearranging :

[tex] (\frac {d^2x} {dt^2} dt) * \frac {dx} {dt} = -kx *dx [/tex]

And the step which I'm not convinced can be done:

[tex]( \int \frac {d^2x} {dt^2} dt) * \frac {dx} {dt} = -k \int x *dx [/tex]

and so we end up with v^2 = -(kx^2)/2 +c which is but an errant 1/2 away from what I need.

I just don't trust integrating a wrt t, while having a v just sitting there as if it were a constant. The only reason that I haven't already disregarded it is that it checks out if you analyse the dimensions...