DISCUSSION
The path variables of radius of curvature ρ and speed v are provided in terms of the distance traveled variable s. Please recall that the time rate of change of speed v_dot that appears in the acceleration vector is (as the name implies) the derivative of v with respect to TIME. Since we know v as a function of position and not time, we need to use the chain rule of derivatives to find the rate of change of speed:  v_dot = dv/dt = (dv/ds)*(ds/dt) = v*dv/ds.

Please ask and answer questions here. Either way, you learn.