This is not the first time I am studying classical mechanics but the idea of reference frames has always confused me, more so after studying a bit of relativity. I'd really like it if you could clear this up for me once and for all.
Consider a frame $S'$ moving at a velocity $v$ w.r.t a frame $S$ at a certain instant, such that $S'$ is also accelerating w.r.t $S$, its acceleration being $a$. Also, let $S$ be a frame that is at rest w.r.t me.
Let $u_x,u_x'$ be the speeds of an object in $S$ and $S'$ respectively. Then obviously $u_x'=u_x-v$. Differentiating this we get $a_x'=a_x-a$. Clearly, acceleration is relative.
If that is the case, then I don't see any point in defining inertial and non-inertial reference frames, because acceleration being equal to zero will be relative, which will obviously mess up all the formulations of the theory.
I've tried again and again to grasp this concept but I feel that things somehow just don't add up. Please help me.