Let be a point at a distance from the center of a circle of radius . The curve traced

Chapter 10, Problem 40

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Let be a point at a distance from the center of a circle of radius . The curve traced out by as the circle rolls along a straight line is called a trochoid. (Think of the motion of a point on a spoke of a bicycle wheel.) The cycloid is the special case of a trochoid with . Using the same parameter as for the cycloid and, assuming the line is the -axis and when is at one of its lowest points, show that parametric equations of the trochoid are Sketch the trochoid for the cases and . Suppose that the position of one particle at time is given by and the position of a second particle is given by (a) Graph the paths of both particles. How many points of intersection are there? (b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place at the same time? If so, find the collision points. (c) Describe what happens if the path of the second particle is given by