A Point on a Bicycle Wheel

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Roll a bicycle wheel along a night road, watch the reflector on its rim trace a cycloid, and differentiate with respect to the parameter θ to get the reflector’s velocity and the slope of the tangent. Find that the reflector stops for an instant when it touches the ground (the cusp) and moves at twice the bike’s speed at the top, then compare it with the smooth curve traced by a point on a spoke.

Key conceptsparameter · derivative of a parametric function · cycloid · slope of the tangent · derivative · velocity · derivatives of trigonometric functions

• Calculus