Owl: Planar motion is a vector story. Position r(t) traces a path. Velocity v = r′ points along the path with speed |v|. Acceleration a = v′ can change speed, direction, or both.
Here's the puzzle that breaks intuition: turn a corner at constant speed — why is acceleration nonzero? Because direction is changing. Tangential acceleration stretches the speedometer; normal (centripetal) acceleration bends the heading.
We'll differentiate vector functions, connect to earlier parametric work, and separate 'speeding up' from 'turning.' Acceleration does not always increase speed.