Conceptual difficulty with with component accelerations in polar coordinates

Question

In kinematics why is the acceleration along the radius not the time derivative of the velocity along that radius?

Answer 1

Mathematically, this is because the directions are time-dependent. When you use Cartesian coordinates, you enjoy the privilege that $\frac{d}{dt} \mathbf{\hat{x}} = \frac{d}{dt} \mathbf{\hat{y}} = \mathbf{0}$, but this isn't true in polar coordinates, for the meaning of "radial direction" and "tangential direction" change from an instant to the next.

Physically, think of centripetal acceleration: to keep a body spinning at constant tangential velocity, you actually have to accelerate it toward the center. After all, the velocity is changing direction, meaning it is not constant, and hence there's got to be some acceleration. This is necessary to counterbalance the body's inertia, which attempts to keep it moving in a straight line.