Sketch the trochoid for the casesand.drdyrdcosxrdsinP0xdrPrdP2ye2txetysec2txcostyt2xye2txe3tyt4xt6yt2xt3rt0yx238402yx2a334.0, 32, 12, 1x2y12433.C1, 5B4, 2A1, 13,12, 7131.628||||CHAPTER 10PARAMETRIC EQUATIONS AND POLAR COORDINATES

LABORATORY PROJECTRUNNING CIRCLES AROUND CIRCLES||||629given by the parametric equationswhereis the acceleration due to gravity (m s ).(a) If a gun is fired withandms, whenwill the bullet hit the ground? How far from the gun willit hit the ground? What is the maximum height reachedby the bullet?05003029.8tyv0sint2tt2xv0cos(b) Use the geometric description of the curve to draw arough sketch of the curve by hand. Check your work byusing the parametric equations to graph the curve.t2y21sintx23cos0t2y12 costx13 sintxOyAPx=2aBa;(b) Use a graphing device to check your answers to part (a).Then graph the path of the projectile for several othervalues of the angleto see where it hits the ground.Summarize your findings.(c) Show that the path is parabolic by eliminating theparameter.;Investigate the family of curves defined by the parametricequations,. How does the shape changeasincreases? Illustrate by graphing several members of thefamily.;48.Theswallowtail catastrophe curvesare defined by the para-metric equations,. Graphseveral of these curves. What features do the curves havein common? How do they change whenincreases?;The curves with equations,arecalledLissajous figures. Investigate how these curves varywhen , , andvary. (Taketo be a positive integer.);50.Investigate the family of curves defined by the parametricequations,, where. Startby lettingbe a positive integer and see what happens to theshape asincreases. Then explore some of the possibilitiesthat occur whenis a fraction.ccc0ysintsinctxcosnnbaybcostxasinnt49.cyct23t4x2ct4t3cyt3ctxt2collision pointsIn other words, are the particles ever at the same place atthe same time? If so, find the collision points.(c) Describe what happens if the path of the second particleis given by46.If a projectile is fired with an initial velocity ofmeters persecond at an angleabove the horizontal and air resistanceis assumed to be negligible, then its position afterseconds isv0x23costy21sint0thypocycloidsandepicycloids,that aregenerated by the motion of a point on a circle that rolls inside or outside another circle.ct47.v05003029.8tyv0sint2tt2xv0cos(b) Use the geometric description of the curve to draw arough sketch of the curve by hand. Check your work byusing the parametric equations to graph the curve.t2y21sintx23cos0t2y12 costx13 sintxOyAPx=2aBa1t;45.Suppose that the position of one particle at timeis given byand the position of a second particle is given by(a) Graph the paths of both particles. How many points ofintersection are there?(b) Are any of these points of intersectioncollision pointsIn other words, are the particles ever at the same place atthe same time? If so, find the collision points.(c) Describe what happens if the path of the second particleis given by46.If a projectile is fired with an initial velocity ofmeters persecond at an angleabove the horizontal and air resistanceis assumed to be negligible, then its position afterseconds isv0x23costy21sint0thypocycloidsandepicycloids,that aregenerated by the motion of a point on a circle that rolls inside or outside another circle.?t20t2y21sintx23cos0t2y12 costx13 sintxOyAPx=2aBattIn this project we investigate families of curves, calledhypocycloidsandepicycloids,that aregenerated by the motion of a point on a circle that rolls inside or outside another circle.

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Term

Spring

Professor

LYLES

Tags

Calculus, Equations, Cartesian Coordinate System, Parametric Equations, Polar Coordinates, Polar coordinate system, Parametric equation, Conic section