I was trying to get the Specific Impulse of the Saturn V engines, hoping for a value in N/kg/s, as I need to know the mass consumption rate per thrust value. Any and all sources give me the value in seconds: 263s, rather than what I'm looking for. Trying to convert from this unit into N/Kg/s leaves me clueless as I end up at m/s. Now I know dividing velocity by acceleration, g, leaves you with seconds, thus having: $$s = \frac{N}{kg/s} = \frac{kg\times m}{s^2} \times \frac{s}{kg}= \frac{m}{s}$$ To finish with units of s: $$s = \frac{m}{s}\times \frac{s^2}{m}$$ Thus, $$I_{sp}\space\left(\frac{N}{kg/s}\right) = g \times I_{sp}(s)$$
But I can't figure out why you must introduce g into the equation as it doesn't appear during the dimensional analysis. I very well could be missing critical information though which is why I'm here.
Additionally different sources give me different formulas for the Isp itself, some saying it's $\frac{F}{\dot{m}}\space$versus$\space\frac{F*\Delta T}{\dot{m}}.$ The NASA webpage for $I_{sp}$ words the formula as $\frac{Total-Impulse}{\dot{m}* g}$, but the final formula shows as $\frac{F}{\dot{m*g}}$.
