Sketch the trochoid for the cases and d r d r y r d cos x r d sin P x d r P r d

Sketch the trochoid for the cases and d r d r y r d

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Sketch the trochoid for the cases and . d r d r y r d cos x r d sin P 0 x d r P r d P 2 y e 2 t x e t y sec 2 t x cos t y t 2 x t y e 2 t x e 3 t y t 4 x t 6 y t 2 x t 3 0 y x 2 3 8 4 0 2 y x 2 a 3 x 2 a 2 y 2 b 2 1 34. 0, 3 2, 1 2, 1 x 2 y 1 2 4 33. C 1, 5 B 4, 2 A 1, 1 3, 1 2, 7 P 2 x 2 , y 2 P 1 x 1 , y 1 0 t 1 y y 1 y 2 y 1 t x x 1 x 2 x 1 t 31. 628 |||| CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
LABORATORY PROJECT RUNNING CIRCLES AROUND CIRCLES |||| 629 given by the parametric equations where is the acceleration due to gravity ( m s ). (a) If a gun is fired with and m s, when will the bullet hit the ground? How far from the gun will it hit the ground? What is the maximum height reached by the bullet? ; (b) Use a graphing device to check your answers to part (a). Then graph the path of the projectile for several other values of the angle to see where it hits the ground. Summarize your findings. (c) Show that the path is parabolic by eliminating the parameter. ; Investigate the family of curves defined by the parametric equations , . How does the shape change as increases? Illustrate by graphing several members of the family. ; 48.The swallowtail catastrophe curvesare defined by the para-metric equations , . Graph several of these curves. What features do the curves have in common? How do they change when increases? 50. Investigate the family of curves defined by the parametric equations , , where . Start by letting be a positive integer and see what happens to the shape as increases. Then explore some of the possibilities that occur when is a fraction. c c c c 0 y sin t sin ct x cos t n n b a y b cos t x a sin nt 49. c y ct 2 3 t 4 x 2 ct 4 t 3 c y t 3 ct x t 2 47. v 0 500 30 2 9.8 t y v 0 sin t 1 2 t t 2 x v 0 cos t (b) Use the geometric description of the curve to draw a rough sketch of the curve by hand. Check your work by using the parametric equations to graph the curve. ; 45. 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 46. If a projectile is fired with an initial velocity of meters per second at an angle above the horizontal and air resistance is assumed to be negligible, then its position after seconds is t v 0 x 2 3 cos t y 2 1 sin t 0 t 2 0 t 2 y 2 1 sin t x 2 3 cos t 0 t 2 y 1 2 cos t x 1 3 sin t t x O y A P x=2a B a In this project we investigate families of curves, called hypocycloids and epicycloids, that are generated by the motion of a point on a circle that rolls inside or outside another circle.