Question 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.
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Differentiate vector-valued position to get velocity and acceleration Relate speed to |v| and connect to parametric calculus Explain nonzero acceleration at constant speed when turning Distinguish tangential vs normal contributions to acceleration
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Constant speed turn: why a ≠ 0? r(t) → v = r′ → a = v′ speed = |v|
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