Falling body equation

Hello,



I don't think I really understand the derivation of an equation of motion for a falling body under the force of earth's gravity... I'm a little ashamed by this. I haven't been able to find someone asking quite the same question as I am about to ask.



I completely understand that the acceleration of an object is the second derivative of its position, and than an accelerated body is not in equilibrium, so ∑F ≠ 0.



I've been able to get by but it's time I truly understand what is going on. From the perspective of differential equations the derivation starts with an expression:



md²s/dt² = -mg. Then masses cancel to get d²s/dt² = -g.



What are we actually saying by equating two forces? I just see ma = -ma. But what does this statement physically mean?



I feel safe asking such a basic question on this forum and I trust that people will only give positive feedback in this situation. If I failed to find an online resource that explains this on the level I am seeking it would not be insulting to simply point me there. Thanks in advance,



Lee