I am trying to convert something from time domain to Laplace then back to time, but it isn't working. Here is a simple way to illustrate, with force (F), mass (m), acceleration (a), velocity (V), and position (y):
$F(t) = m a(t)$
$F(s) = m a(s)$
$F(s) = m (sV(s) - v0)$
$F(s) = m (s^{2}y(s) - sy(0) - y'(0))$, if y0=0, then:
$F(s) = ms^{2}y(s)-mv(0)$
$y(s) = \frac{F(s)+mv(0)}{ms^{2}}$
Then given: $\frac{1}{s^{2}} = t$ from here:
$y(t) = \frac{(F(t)+mv(0)) * t}{m}$
Only that doesn't actually make sense, because then you are saying:
$displacement = \frac{(force + momentum) * time}{mass}$
The units don't work out.
So where was the error in the above to explain the problem?