but what does, say, a centripetal acceleration of 1m/s² means?
It means that the net force per unit mass must act on the object of magnitude 1 Newton/kilogram toward the center of the circular motion, if it is to travel in a circle at constant speed.
Ok, if you aren't ready to think about forces, it means that the velocity vector is changing at the rate given by the acceleration vector.
The acceleration vector points inward, and constantly drags the velocity vector to follow it, but never changes its speed (only if there is tangential acceleration does speed change).
Where does the formula come from? Let's derive it.
Consider a body on an x-y coordinate system with speed v traveling in the +y direction and it is at a location of x=r, y=0. It is circling around the origin as you would likely guess.
At this location its velocity vector is V = <0, V>
And its displacement vector is D = <r, 0>
Now, as a function of time, it is traveling in a circle.
Its displacement is after traveling an angle of theta:
D = <r*cos(theta), r*sin(theta)>
Recall how nicely the radian is defined, 1 radius wrapped around the circle?
d = theta*r
where d is arc-length traveled.
D = <r*cos(d/r), r*sin(d/r)>
Take the derivative to find the velocity vector:
V = dD/dt = <-v*sin(v*t/r), v*cos(v*t/r)>
Take derivative again to find acceleration vector:
A = dV/dt = <-v*v/r*cos(v*t/r), -v*v/r*sin(v*t/r)>
a = |A|
a = sqrt((-v*v/r*cos(v*t/r))^2 + (-v*v/r*sin(v*t/r))^2)
a = sqrt((v*v/r)^2*(cos(v*t/r))^2 + (sin(v*t/r))^2))
a = (v^2/r)*sqrt((cos(v*t/r))^2 + (sin(v*t/r))^2))
a = (v^2/r)*sqrt(1)
a = v^2/r