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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
82E83ERelationship between r and r 78. Consider the circle r(t) = a cos t, a sin t, for 0 t 2, where a is a positive real number. Compute r and show that it is orthogonal to r for all t.Relationship between r and r 79. Consider the parabola r(t) = at2 + 1, t, for t , where a is a positive real number. Find all points on the parabola at which r and r are orthogonal.86ERelationship between r and r 81. Consider the helix r(t) = cos t, sin t, t, for t . Find all points on the helix at which r and r are orthogonal.Relationship between r and r 82. Consider the ellipse r(t) = 2 cos t, 8 sin t, 0, for 0 t 2. Find all points on the ellipse at which r and r are orthogonal.Relationship between r and r 83. Give two families of curves in 3 for which r and r are parallel for all t in the domain.Motion on a sphere Prove that r describes a curve that lies on the surface of a sphere centered at the origin (x2 + y2 + z2 = a2 with a 0) if and only if r and r are orthogonal at all points of the curve.Vectors r and r for lines a. If r(t) = at, bt, ct with a, b, c 0, 0, 0, show that the angle between r and r is constant for all t 0. b. If r(t) = x0 + at, y0 + bt, z0 + ct, where x0, y0, and z0 are not all zero, show that the angle between r and r varies with t. c. Explain the results of parts (a) and (b) geometrically.Proof of Sum Rule By expressing u and v in terms of their components, prove that ddt(u(t)+v(t))=u(t)+v(t).Proof of Product Rule By expressing u in terms of its components, prove that ddt(f(t)u(t))=f(t)u(t)+f(t)u(t).94ECusps and noncusps a. Graph the curve r(t) = t3, t3. Show that r(0) = 0 and the curve does not have a cusp at t = 0. Explain. b. Graph the curve r(t) = t3, t2. Show that r(0) = 0 and the curve has a cusp at t = 0. Explain. c. The functions r(t) = t, t2 and p(t) = t2, t4 both satisfy y = x2. Explain how the curves they parameterize are different. d. Consider the curve r(t) = tm, tn, where m 1 and n 1 are integers with no common factors. Is it true that the curve has a cusp at t = 0 if one (not both) of m and n is even? Explain.Given r(t)=t,t2,t3, find v(t) and a(t).Find the functions that give the speed of the two objects in Example 2, for t 0 (Corresponding to the graphs in Figure 14.15). Example 2 Comparing trajectories Consider the trajectories described by the position functions r(t)t,t24,t348, for t 0, and R(t)t2,t44,t648, for t 0, Where t is measured in the same time units for both functions. a. Graph and compare the trajectories using a graphing utility. b. Find the velocity vectors associated with the position functions. Figure 14.153QC4QC5QCGiven the position function r of a moving object, explain how to find the velocity, speed, and acceleration of the object.What is the relationship between the position and velocity vectors for motion on a circle?Write Newtons Second Law of Motion in vector form.Write Newtons Second Law of Motion for three-dimensional motion with only the gravitational force (acting in the z-direction).Given the acceleration of an object and its initial velocity, how do you find the velocity of the object, for t 0?Given the velocity of an object and its initial position, how do you find the position of the object for t 0?The velocity of a moving object, for t 0, is r(t)=60,9632t ft/s. a. When is the vertical component of velocity of the object equal to 0? b. Find r(t) if r(0)=0,3.A baseball is hit 2 feet above home plate, and the position of the ball t seconds later is r(t)=40t,16t2+31t+2 ft. Find each of the following values. a. The time of flight of the baseball b. The range of the baseballVelocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 7. r(t) = 3t2 + 1, 4t2 + 3, for t 0Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 8. r(t)=52t2+3,6t2+10,fort0Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 9. r(t) = 2 + 2t, 1 4t, for t 0Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 10. r(t) = 1 t2, 3 + 2t3, for t 0Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 11. r(t) = 8 sin t, 8 cos t, for t 2Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 12. r(t) =3 cos t, 4 sin t, for 0 t 2Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 13. r(t)=t2+3,t2+10,12t2,fort0Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 14. r(t) = 2e2t + 1, e2t 1, 2e2t 10, for t 0Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 15. r(t) = 3 + t, 2 4t, 1 + 6t, for t 0Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 16. r(t) = 3 sin t, 5 cos t, 4 sin t, for 0 t 2Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 17. r(t) = l, t2, et, for t 0Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 18. r(t) = 13 cos 2t, 12 sin 2t, 5 sin 2t, for 0 tComparing trajectories Consider the following position functions r and R for two objects. a. Find the interval [c, d] over which the R trajectory is the same as the r trajectory over [a, b]. b. Find the velocity for both objects. c. Graph the speed of the two objects over the intervals [a, b] and [c, d], respectively. 19. r(t) = t, t2, [a, b] = [0, 2], R(t) = 2t, 4t2) on [c, d]Comparing trajectories Consider the following position functions r and R for two objects. a. Find the interval [c, d] over which the R trajectory is the same as the r trajectory over [a, b]. b. Find the velocity for both objects. c. Graph the speed of the two objects over the intervals [a, b] and [c, d], respectively. 20. r(t) = (1 + 3t, 2 + 4t, [a, b] = [0, 6], R(t) = (1 + 9t, 2 + 12t) on [c, d]Comparing trajectories Consider the following position functions r and R for two objects. a. Find the interval [c, d] over which the R trajectory is the same as the r trajectory over [a, b]. b. Find the velocity for both objects. c. Graph the speed of the two objects over the intervals [a, b] and [c, d], respectively. 21. r(t) = cos t, 4 sin t, [a, b] = [0, 2], R(t) = cos 3t, 4 sin 3t on [c, d]Comparing trajectories Consider the following position functions r and R for two objects. a. Find the interval [c, d] over which the R trajectory is the same as the r trajectory over [a, b]. b. Find the velocity for both objects. c. Graph the speed of the two objects over the intervals [a, b] and [c, d], respectively. 22. r(t) = 2 et, 4 et,[a, b] = [0, ln 10], R(t) = 2 t, 4 1/t) on [c, d]Comparing trajectories Consider the following position functions r and R for two objects. a. Find the interval [c, d] over which the R trajectory is the same as the r trajectory over [a, b]. b. Find the velocity for both objects. c. Graph the speed of the two objects over the intervals [a, b] and [c, d], respectively. 23. r(t) = 4 + t2, 3 2t4, 1 + 3t6, [a, b] = [0, 6], R(t) = 4 + ln t, 3 2 ln2 t, 1 + 3 ln3 t on [c, d] For graphing, let c = 1 and d = 20.Comparing trajectories Consider the following position functions r and R for two objects. a. Find the interval [c, d] over which the R trajectory is the same as the r trajectory over [a, b]. b. Find the velocity for both objects. c. Graph the speed of the two objects over the intervals [a, b] and [c, d], respectively. 24. r(t) = 2 cos 2t, 2sin2t,2sin2t, [a, b] = [0, ], R(t) = 2 cos 4t, 2sin4t, 2sin4t on [c, d]27ECarnival rides 28. Suppose the carnival ride in Exercise 27 is modified so that Andreas position P (in ft) at time t (in s) is r(t)=20cost+10cos5t,20sint+10sin5t,5sin2t. a. Describe how this carnival ride differs from the ride in Exercise 27. b. Find the speed function |v(t)| = (t) and plot its graph. c. Find Andreas maximum and minimum speeds. 27 Consider a carnival ride where Andrea is at point P that moves counterclockwise around a circle centered at C while the arm, represented by the line segment from the origin O to point C, moves counterclockwise about the origin (see figure). Andrea's position (in feet) at time t (in seconds) is r(t)=20cost+10cos5t,20sint+10sin5t a. Plot a graph of r(t), for 0 t 2. b. Find the velocity v(t). c. Show that the speed v(t)=(t)=1029+20cos4tand plot the speed, for 0 t 2.(Hint: Use the identity sin mx sin nx + cos mx cos nx = cos((m n)x).) d. Determine Andreas maximum and minimum speeds.Trajectories on circles and spheres Determine whether the following trajectories lie on a circle in 2 or sphere in 3 centered at the origin. If so, find the radius of the circle or sphere and show that the position vector and the velocity vector are everywhere orthogonal. 25. r(t) = 8 cos 2t, 8 sin 2t, for 0 t30ETrajectories on circles and spheres Determine whether the following trajectories lie on a circle in 2 or sphere in 3 centered at the origin. If so, find the radius of the circle or sphere and show that the position vector and the velocity vector are everywhere orthogonal. 27. r(t)=sint+3cost,3sintcost, for 0 t 2Trajectories on circles and spheres Determine whether the following trajectories lie on a circle in 2 or sphere in 3 centered at the origin. If so, find the radius of the circle or sphere and show that the position vector and the velocity vector are everywhere orthogonal. 28. r(t) = 3 sin t, 5 cos t, 4 sin t, for 0 t 2Path on a sphere show that the following trajectories lie on a sphere centered at the origin, and find the radius of the sphere. 33. r(t)=5sint1+sin22t,5cost1+sin22t,5sin2t1+sin22t, for 0 t 2Path on a sphere show that the following trajectories lie on a sphere centered at the origin, and find the radius of the sphere. 34.r(t)=4cost4+t2,2t4+t2,4sint4+t2 , for 0 t 4Solving equations of motion Given an acceleration vector, initial velocity u0, v0, and initial position x0, y0, find the velocity and position vectors, for t 0. 31. a(t)= 0, 1, u0, v0 = 2, 3, x0, y0 = 0, 0Solving equations of motion Given an acceleration vector, initial velocity u0, v0, and initial position x0, y0, find the velocity and position vectors, for t 0. 32. a(t) = 1, 2, u0, v0 = 1, 1, x0, y0 = 2, 3Solving equations of motion Given an acceleration vector, initial velocity u0, v0, and initial position x0, y0, find the velocity and position vectors, for t 0. 33. a(t) = 0, 10, u0, v0 = 0, 5, x0, y0 = 1, 1Solving equations of motion Given an acceleration vector, initial velocity u0, v0, and initial position x0, y0, find the velocity and position vectors, for t 0. 34. a(t) = 1, t, u0, v0 = 2, 1, x0, y0 = 0, 8Solving equations of motion Given an acceleration vector, initial velocity u0, v0, and initial position x0, y0, find the velocity and position vectors, for t 0. 35. a(t) = cos t, 2 sin t, u0, v0 = 0, 1, x0, y0 = 1, 0Solving equations of motion Given an acceleration vector, initial velocity u0, v0, and initial position x0, y0, find the velocity and position vectors, for t 0. 36. a(t) = et, 1, u0, v0 = 1, 0, x0, y0 = 0, 0Two-dimensional motion Consider the motion of the following objects. Assume the x-axis is horizontal, the positive y-axis is vertical, the ground is horizontal, and only the gravitational force acts on the object. a. Find the velocity and position vectors, for t 0. b. Graph the trajectory. c. Determine the time of flight and range of the object. d. Determine the maximum height of the object. 37. A soccer ball has an initial position x0, y0 = 0, 0 when it is kicked with an initial velocity of u0, v0 = 30, 6 m/s.Two-dimensional motion Consider the motion of the following objects. Assume the x-axis is horizontal, the positive y-axis is vertical, the ground is horizontal, and only the gravitational force acts on the object. a. Find the velocity and position vectors, for t 0. b. Graph the trajectory. c. Determine the time of flight and range of the object. d. Determine the maximum height of the object. 38. A golf ball has an initial position x0, y0 = 0, 0 when it is hit at an angle of 30 with an initial speed of 150 ft/s.Two-dimensional motion Consider the motion of the following objects. Assume the x-axis is horizontal, the positive y-axis is vertical, the ground is horizontal, and only the gravitational force acts on the object. a. Find the velocity and position vectors, for t 0. b. Graph the trajectory. c. Determine the time of flight and range of the object. d. Determine the maximum height of the object. 39. A baseball has an initial position (in feet) of x0, y0 = 0, 6 when it is thrown with an initial velocity of u0, v0 = 80, 10 ft/s.Two-dimensional motion Consider the motion of the following objects. Assume the x-axis is horizontal, the positive y-axis is vertical, the ground is horizontal, and only the gravitational force acts on the object. a. Find the velocity and position vectors, for t 0. b. Graph the trajectory. c. Determine the time of flight and range of the object. d. Determine the maximum height of the object. 40. A baseball is thrown horizontally from a height of 10 ft above the ground with a speed of 132 ft/s.Two-dimensional motion Consider the motion of the following objects. Assume the x-axis is horizontal, the positive y-axis is vertical, the ground is horizontal, and only the gravitational force acts on the object. a. Find the velocity and position vectors, for t 0. b. Graph the trajectory. c. Determine the time of flight and range of the object. d. Determine the maximum height of the object. 41. A projectile is launched from a platform 20 ft above the ground at an angle of 60 above the horizontal with a speed of 250 ft/s. Assume the origin is at the base of the platform.Two-dimensional motion Consider the motion of the following objects. Assume the x-axis is horizontal, the positive y-axis is vertical, the ground is horizontal, and only the gravitational force acts on the object. a. Find the velocity and position vectors, for t 0. b. Graph the trajectory. c. Determine the time of flight and range of the object. d. Determine the maximum height of the object. 42. A rock is thrown from the edge of a vertical cliff 40 m above the ground at an angle of 45 above the horizontal with a speed of 102m/s. Assume the origin is at the foot of the cliff.Solving equations of motion Given an acceleration vector, initial velocity u0, v0, w0, and initial position x0, y0, z0, find the velocity and position vectors, for t 0. 43. a(t) = 0, 0, 10, u0, v0, w0 = 1, 5, 0, x0, y0, z0 = 0, 5, 0Solving equations of motion Given an acceleration vector, initial velocity u0, v0, w0, and initial position x0, y0, z0, find the velocity and position vectors, for t 0. 44. a(t) = 1, t, 4t, u0, v0, w0 = 20, 0, 0, x0, y0, z0 = 0, 0, 0Solving equations of motion Given an acceleration vector, initial velocity u0, v0, w0, and initial position x0, y0, z0, find the velocity and position vectors, for t 0. 45. a(t) = sin t, cos t, 1, u0, v0, w0 = 0, 2, 0, x0, y0, z0 = 0, 0, 050EThree-dimensional motion Consider the motion of the following objects. Assume the x-axis points east, the y-axis points north, the positive z-axis is vertical and opposite g, the ground is horizontal, and only the gravitational force acts on the object unless otherwise stated. a. Find the velocity and position vectors, for t 0. b. Make a sketch of the trajectory. c. Determine the time of flight and range of the object. d. Determine the maximum height of the object. 47. A bullet is fired from a rifle 1 m above the ground in a northeast direction. The initial velocity of the bullet is 200, 200, 0 m/s.Three-dimensional motion Consider the motion of the following objects. Assume the x-axis points east, the y-axis points north, the positive z-axis is vertical and opposite g, the ground is horizontal, and only the gravitational force acts on the object unless otherwise stated. a. Find the velocity and position vectors, for t 0. b. Make a sketch of the trajectory. c. Determine the time of flight and range of the object. d. Determine the maximum height of the object. 48. A golf ball is hit east down a fairway with an initial velocity of 50, 0, 30 m/s. A crosswind blowing to the south produces an acceleration of the ball of 0.8 m/s2.Three-dimensional motion Consider the motion of the following objects. Assume the x-axis points east, the y-axis points north, the positive z-axis is vertical and opposite g, the ground is horizontal, and only the gravitational force acts on the object unless otherwise stated. a. Find the velocity and position vectors, for t 0. b. Make a sketch of the trajectory. c. Determine the time of flight and range of the object. d. Determine the maximum height of the object. 49. A baseball is hit 3 ft above home plate with an initial velocity of 60, 80, 80 ft/s. The spin on the baseball produces a horizontal acceleration of the ball of 10 ft/s2 in the eastward direction.Three-dimensional motion Consider the motion of the following objects. Assume the x-axis points east, the y-axis points north, the positive z-axis is vertical and opposite g, the ground is horizontal, and only the gravitational force acts on the object unless otherwise stated. a. Find the velocity and position vectors, for t 0. b. Make a sketch of the trajectory. c. Determine the time of flight and range of the object. d. Determine the maximum height of the object. 50. A baseball is hit 3 ft above home plate with an initial velocity of 30, 30, 80 ft/s. The spin on the baseball produces a horizontal acceleration of the ball of 5 ft/s2 in the northward direction.Three-dimensional motion Consider the motion of the following objects. Assume the x-axis points east, the y-axis points north, the positive z-axis is vertical and opposite g, the ground is horizontal, and only the gravitational force acts on the object unless otherwise stated. a. Find the velocity and position vectors, for t 0. b. Make a sketch of the trajectory. c. Determine the time of flight and range of the object. d. Determine the maximum height of the object. 51. A small rocket is fired from a launch pad 10 m above the ground with an initial velocity, in m/s, of 300, 400, 500. A crosswind blowing to the north produces an acceleration of the rocket of 2.5 m/s2.56E57ETrajectory properties Find the time of flight, range, and maximum height of the following two-dimensional trajectories, assuming no forces other than gravity. In each case, the initial position is 0, 0 and the initial velocity is v0 = u0, v0. 54. u0, v0 = 10, 20 ft/sTrajectory properties Find the time of flight, range, and maximum height of the following two-dimensional trajectories, assuming no forces other than gravity. In each case, the initial position is 0, 0 and the initial velocity is v0 = u0, v0. 55. Initial speed |v0| = 150 m/s, launch angle 30Trajectory properties Find the time of flight, range, and maximum height of the following two-dimensional trajectories, assuming no forces other than gravity. In each case, the initial position is 0, 0 and the initial velocity is v0 = u0, v0. 56. u0, v0 = 40, 80 m/sTrajectory properties Find the time of flight, range, and maximum height of the following two-dimensional trajectories, assuming no forces other than gravity. In each case, the initial position is 0, 0 and the initial velocity is v0 = u0, v0. 57. Initial speed |v0| = 400 ft/s, launch angle = 60Motion on the moon The acceleration due to gravity on the moon is approximately g/6 (one-sixth its value on Earth). Compare the time of flight, range, and maximum height of a projectile on the moon with the corresponding values on Earth.Firing angles A projectile is fired over horizontal ground from the origin with an initial speed of 60 m/s. What firing angles produce a range of 300 m?64ESpeed on an ellipse An object moves along an ellipse given by the function r(t) = a cos t, b sin t, for 0 t 2, where a 0 and b 0. a. Find the velocity and speed of the object in terms of a and b, for 0 t 2. b. With a = 1 and b = 6, graph the speed function, for 0 t 2. Mark the points on the trajectory at which the speed is a minimum and a maximum. c. Is it true that the object speeds up along the flattest (straightest) parts of the trajectory and slows down where the curves are sharpest? d. For general a and b, find the ratio of the maximum speed to the minimum speed on the ellipse (in terms of a and b).Golf shot A golfer stands 390 ft (130 yd) horizontally from the hole and 40 ft below the hole (see figure). Assuming the ball is hit with an initial speed of 150 ft/s, at what angle(s) should it be hit to land in the hole? Assume that the path of the ball lies in a plane.Another golf shot A golfer stands 420 ft (140 yd) horizontally from the hole and 50 ft above the hole (see figure). Assuming the ball is hit with an initial speed of 120 ft/s, at what angle(s) should it be hit to land in the hole? Assume that the path of the ball lies in a plane.68EInitial speed of a golf shot A golfer stands 420 ft horizontally from the hole and 50 ft above the hole (see figure for Exercise 67). If the ball leaves the ground at an initial angle of 30 with the horizontal, with what initial speed should it be hit to land in the hole? Another golf shot A golfer stands 420 ft (140 yd) horizontally from the hole and 50 ft above the hole (see figure). Assuming the ball is hit with an initial speed of 120 ft/s, at what angle(s) should it be hit to land in the hole? Assume the path of the ball lies in a plane.Ski jump The lip of a ski jump is 8 m above the outrun that is sloped at an angle of 30 to the horizontal (see figure). a. If the initial velocity of a ski jumper at the lip of the jump is 40, 0 m/s, what is the length of the jump (distance from the origin to the landing point)? Assume only gravity affects the motion. b. Assume that air resistance produces a constant horizontal acceleration of 0.15 m/s2 opposing the motion. What is the length of the jump? c. Suppose that the takeoff ramp is tilted upward at an angle of , so that the skiers initial velocity is 40 cos , sin m/s. What value of maximizes the length of the jump? Express your answer in degrees and neglect air resistance.Designing a baseball pitch A baseball leaves the hand of a pitcher 6 vertical feet above and 60 horizontal feet from home plate. Assume that the coordinate axes are oriented as shown in the figure. a. In the absence of all forces except gravity, assume that a pitch is thrown with an initial velocity of 130, 0, 3 ft/s (about 90 mi/hr). How far above the ground is the ball when it crosses home plate and how long does it take the pitch to arrive? b. What vertical velocity component should the pitcher use so that the pitch crosses home plate exactly 3 ft above the ground? c. A simple model to describe the curve of a baseball assumes that the spin of the ball produces a constant sideways acceleration (in the y-direction) of c ft/s2. Assume a pitcher throws a curve ball with c = 8 ft/s2 (one-fourth the acceleration of gravity). How far does the ball move in the y-direction by the time it reaches home plate, assuming an initial velocity of 130, 0, 3 ft/s? d. In part (c), does the ball curve more in the first half of its trip to the plate or in the second half? How does this fact affect the batter? e. Suppose the pitcher releases the ball from an initial position of 0, 3, 6 with initial velocity 130, 0, 3. What value of the spin parameter c is needed to put the ball over home plate passing through the point (60, 0, 3)?Parabolic trajectories Show that the two-dimensional trajectory x(t) = u0 t + x0 and y(t)=gt22+v0t+y0, for 0 t T, of an object moving in a gravitational field is a segment of a parabola for some value of T 0. Find T such that y(T) = 0.73EA race Two people travel from P(4, 0) to Q(4, 0) along the paths given by r(t)=4cos(t/8),4sin(t/8)andR(t)=4t,(4t)216. a. Graph both paths between P and Q. b. Graph the speeds of both people between P and Q. c. Who arrives at Q first?Circular motion Consider an object moving along the circular trajectory r(t) = A cos t, A sin t, where A and are constants. a. Over what time interval [0, T] does the object traverse the circle once? b. Find the velocity and speed of the object. Is the velocity constant in either direction or magnitude? Is the speed constant? c. Find the acceleration of the object. d. How are the position and velocity related? How are the position and acceleration related? e. Sketch the position, velocity, and acceleration vectors at four different points on the trajectory with A = = 1.76EA circular trajectory An object moves clockwise around a circle centered at the origin with radius 5 m beginning at the point (0, 5). a. Find a position function r that describes the motion if the object moves with a constant speed, completing 1 lap every 12 s. b. Find a position function r that describes the motion if it occurs with speed et.78ETilted ellipse Consider the curve r(t) = cos t, sin t, c sin t, for 0 t 2, where c is a real number. Assuming the curve lies in a plane, prove that the curve is an ellipse in that plane.Equal area property Consider the ellipse r(t) = a cos t, b sin t, for 0 t 2, where a and b are real numbers. Let be the angle between the position vector and the x-axis. a. Show that tan = (b/a) tan t. b. Find (t). c. Recall that the area bounded by the polar curve r = f() on the interval [0, ] is A()=120(f(u))2du. Letting f((t)) = |r((t))|, show that A(t)=12ab. d. Conclude that as an object moves around the ellipse, it sweeps out equal areas in equal times.Another property of constant | r | motion Suppose an object moves on the surface of a sphere with |r(t)| constant for all t. Show that r(t) and a(t) = r(t) satisfy r(t) a(t) = |v(t)|2.82ENonuniform straight-line motion Consider the motion of an object given by the position function r(t)=f(t)a,b,c+x0,y0,z0,fort0, where a, b, c, x0, y0, and z0 are constants, and f is a differentiable scalar function, for t 0. a. Explain why this function describes motion along a line. b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?What does the arc length formula give for the length of the line r(t) = (2t, t, 2t), for 0 t 3?Consider the portion of a circle r(t) = (cos t, sin t), for a t b. Show that the arc length of the curve is b a.3QCFind the length of the line given by r(t) = t, 2t, for a t b.Explain how to find the length of the curve r(t) = f(t), g(t), h(t), for a t b.Express the arc length of a curve in terms of the speed of an object moving along the curve.Suppose an object moves in space with the position function r(t) = x(t), y(t), z(t). Write the integral that gives the distance it travels between t = a and t = b.An object moves on a trajectory given by r(t) = 10 cos 2t, 10 sin 2t, for 0 t . How far does it travel?Use calculus to find the length of the line segment r(t) = (t, 8t, 4t) , for 0 t 2. Verify your answer without using calculus.Explain what it means for a curve to be parameterized by its arc length.Is the curve r(t) = cos t, sin t parameterized by its arc length? Explain.Arc length calculations Find the length of he following two- and three-dimensional curves. 9. r(t) = 3t2 1, 4t2 + 5, for 0 t 1Arc length calculations Find the length of the following two- and three-dimensional curves. 10. r(t) = 3t 1, 4t + 5, t, for 0 t 1Arc length calculations Find the length of the following two- and three-dimensional curves. 11. r(t) = 3 cos t, 3 sin t, for 0 tArc length calculations Find the length of the following two- and three-dimensional curves. 12. r(t) = 4 cos 3t, 4 sin 3t, for 0 t 2/313EArc length calculations Find the length of the following two- and three-dimensional curves. 14. r(t) = cos t + sin t, cos t sin t, for 0 t 2Arc length calculations Find the length of the following two- and three-dimensional curves. 15. r(t) = 2 + 3t, 1 4t, 4 + 3t, for 1 t 616EArc length calculations Find the length of the following two- and three-dimensional curves. 17. r(t) = t, 8 sin t, 8 cos t, for 0 t 4Arc length calculations Find the length of the following two- and three-dimensional curves. 18. r(t) = t2/2, (2t + 1)3/2/3, for 0 t 2Arc length calculations Find the length of the following two- and three-dimensional curves. 19. r(t) = e2t, 2e2t + 5, 2e2t 20, for 0 t ln 2Arc length calculations Find the length of the following two- and three-dimensional curves. 20. r(t) = t2, t3, for 0 t 4Arc length calculations Find the length of the following two- and three-dimensional curves. 21. r(t) = cos3 t, sin3 t, for 0 t /2Arc length calculations Find the length of the following two- and three-dimensional curves. 22. r(t) = 3 cos t, 4 cos t, 5 sin t, for 0 t 2Speed and arc length For the following trajectories, find the speed associated with the trajectory and then find the length of the trajectory on the given interval. 23. r(t) = 2t3, t3, 5t3, for 0 t 4Speed and arc length For the following trajectories, find the speed associated with the trajectory and then find the length of the trajectory on the given interval. 24. r(t) = 5 cos t2, 5 sin t2, 12t2, for 0 t 2Speed and arc length For the following trajectories, find the speed associated with the trajectory and then find the length of the trajectory on the given interval. 25. r(t) = 13 sin 2t, 12 cos 2t, 5 cos 2t, for 0 tSpeed and arc length For the following trajectories, find the speed associated with the trajectory and then find the length of the trajectory on the given interval. 26. r(t) = et sin t, et cos t, et, for 0 t ln 2Speed of Earth Verify that the length of one orbit of Earth is approximately 6.280 AU (see Table 14.1). Then determine the average speed of Earth relative to the sun in miles per hour. (Hint: It takes Earth 365.25 days to orbit the sun.) Table 14.1Speed of Jupiter Verify that the length of one orbit of Jupiter is approximately 32.616 AU (see Table 14.1). Then determine the average speed of Jupiter relative to the sun in miles per hour. (Hint: It takes Jupiter 11.8618 Earth years to orbit the sun.) Table 14.1Arc length approximations Use a calculator to approximate the length of the following curves. In each case, simplify the arc length integral as much as possible before finding an approximation. 27. r(t) = 2 cos t, 4 sin t, for 0 t 230EArc length approximations Use a calculator to approximate the length of the following curves. In each case, simplify the arc length integral as much as possible before finding an approximation. 29. r(t) = t, 4t2,10, for 2 t 232E33EArc length parameterization Determine whether the following curves use arc length as a parameter. If not, find a description that uses arc length as a parameter. 42. r(t)=t3,t3,t3, for 0 t 10Arc length parameterization Determine whether the following curves use arc length as a parameter. If not, find a description that uses arc length as a parameter. 43. r(t) = t, 2t, for 0 t 3Arc length parameterization Determine whether the following curves use arc length as a parameter. If not, find a description that uses arc length as a parameter. 44. r(t) = t + 1, 2t 3, 6t, for 0 t 1037E38E39EArc length parameterization Determine whether the following curves use arc length as a parameter. If not, find a description that uses arc length as a parameter. 48. r(t) = (t2, 2t2, 4t2), for 1 t 4Arc length parameterization Determine whether the following curves use arc length as a parameter. If not, find a description that uses arc length as a parameter. 49. r(t) = et, et, et, for t 0Arc length parameterization Determine whether the following curves use arc length as a parameter. If not, find a description that uses arc length as a parameter. 50. r(t)=cost2,cost2,sint, for 0 t 10Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If an object moves on a trajectory with constant speed S over a time interval a t b, then the length of the trajectory is S(b a). b. The curves defined by r(t) = f(t), g(t) and R(t) = g(t), f(t) have the same length over the interval [a, b]. c. The curve r(t) = f(t), g(t), for 0 a t b, and the curve R(t) = f(t2), g(t2), for atb, have the same length. d. The curve r(t) = t, t2, 3t2), for 1 t 4, is parameterized by arc length.Length of a line segment Consider the line segment joining the points P(x0, y0, z0) and Q(x1, y1, z1). a. Find a parametric description of the line segment PQ. b. Use the arc length formula to find the length of PQ. c. Use geometry (distance formula) to verify the result of part (b).Tilted circles Let the curve C be described by r(t) = a cos t, b sin t, c sin t, where a, b, and c are real positive numbers. a. Assume that C lies in a plane. Show that C is a circle centered at the origin provided a2 = b2 + c2. b. Find the arc length of the circle. c. Assuming that the curve lies in a plane, find the conditions for which r(t) = a cos t + b sin t, c cos t + d sin t, e cos t + f sin t describes a circle. Then find its arc length.46E47EToroidal magnetic field A circle of radius a that is centered at (A, 0) is revolved about the y-axis to create a torus (assume a < A). When current flows through a copper wire that is wrapped around this torus, a magnetic field is created and the strength of this field depends on the amount of copper wire used. If the wire is wrapped evenly around the torus a total of k times, the shape of the wire is modeled by the function
r(t) =
for 0 ≤ t ≤ 2π. Determine the amount of copper required if A = 4 in, a = 1 in, and k = 35.
Projectile trajectories A projectile (such as a baseball or a cannonball) launched from the origin with an initial horizontal velocity u0 and an initial vertical velocity v0 moves in a parabolic trajectory given by x=u0t,y=12gt2+v0t,fort0 where air resistance is neglected and g 9.8 m/s2 is the acceleration due to gravity (see Section 11.7). a. Let u0 = 20 m/s and v0 = 25 m/s. Assuming the projectile is launched over horizontal ground, at what time does it return to Earth? b. Find the integral that gives the length of the trajectory from launch to landing. c. Evaluate the integral in part (b) by first making the change of variables u = gt + v0. The resulting integral is evaluated either by making a second change of variables or by using a calculator. What is the length of the trajectory? d. How far does the projectile land from its launch site?Variable speed on a circle Consider a particle that moves in a plane according to the equations x = sin t2 and y = cos t2 with a starting position (0, 1) at t = 0 a. Describe the path of the particle, including the time required to return to the starting position. b. What is the length of the path in part (a)? c. Describe how the motion of this particle differs from the motion described by the equations x = sin t and y = cos t. d. Consider the motion described by x = sin tn and y = cos tn, where n is a positive integer. Describe the path of the particle, including the time required to return to the starting position. e. What is the length of the path in part (d) for any positive integer n? f. If you were watching a race on a circular path between two runners, one moving according to x = sin t and y = cos t and one according to x = sin t2 and y = cos t2, who would win and when would one runner pass the other?Arc length parameterization Prove that the line r(t) = x0 + at, y0 + bt, z0 + ct is parameterized by arc length provided a2 + b2 + c2 = 1.Arc length parameterization Prove that the curve r(t) = a cos t, b sin t, c sin t is parameterized by arc length provided a2 = b2 + c2 = 1.53EChange of variables Consider the parameterized curves r(t) = f(t), g(t), h(t) and R(t) = f(u(t)), g(u(t)), h(u(t)), where f, g, h, and u are continuously differentiable functions and u has an inverse on [a, b]. a. Show that the curve generated by r on the interval a t b is the same as the curve generated by R on u1 (a) t u1(b) (or u1(b) t u1(a)). b. Show that the lengths of the two curves are equal. (Hint: Use the Chain Rule and a change of variables in the arc length integral for the curve generated by R.)What is the curvature of the circle r() = 3sin,3cos?Use the alternative curvature formula to compute the curvature of the curve r(t) = t2,10,103QC4QC5QC6QC7QCWhat is the curvature of a straight line?Explain the meaning of the curvature of a curve. Is it a scalar function or a vector function?Give a practical formula for computing the curvature.Interpret the principal unit normal vector of a curve. Is it a scalar function or a vector function?Give a practical formula for computing the principal unit normal vector.Explain how to decompose the acceleration vector of a moving object into its tangential and normal components.Explain how the vectors T, N, and B are related geometrically.How do you compute B?Give a geometrical interpretation of the torsion.How do you compute the torsion?Curvature Find the unit tangent vector T and the curvature for the following parameterized curves. 11. r(t) = 2t + 1, 4t 5, 6t + 12Curvature Find the unit tangent vector T and the curvature for the following parameterized curves. 12. r(t) = 2 cos t, 2 sin tCurvature Find the unit tangent vector T and the curvature for the following parameterized curves. 13. r(t) = 2t, 4 sin t, 4 cos tCurvature Find the unit tangent vector T and the curvature for the following parameterized curves. 14. r(t) = cos t2, sin t2Curvature Find the unit tangent vector T and the curvature for the following parameterized curves. 15. r(t)=3sint,sint,2costCurvature Find the unit tangent vector T and the curvature for the following parameterized curves. 16. r(t) = t, ln cos tCurvature Find the unit tangent vector T and the curvature for the following parameterized curves. 17. r(t) = t, 2t2Curvature Find the unit tangent vector T and the curvature for the following parameterized curves. 18. r(t) = cos3 t, sin3 tCurvature Find the unit tangent vector T and the curvature for the following parameterized curves. 19. r(t)=0tcos(u2/2)du,0tsin(u2/2)du,,t020EAlternative curvature formula Use the alternative curvature formula = |v a|/|v|3 to find the curvature of the following parameterized curves. 21. r(t) = 3 cos t, 3 sin t, 0Alternative curvature formula Use the alternative curvature formula = |v a|/|v|3 to find the curvature of the following parameterized curves. 22. r(t) = 4t, 3 sin t, 3 cos tAlternative curvature formula Use the alternative curvature formula = |v a|/|v|3 to find the curvature of the following parameterized curves. 23. r(t) = 4 + t2, t, 0Alternative curvature formula Use the alternative curvature formula = |v a|/|v|3 to find the curvature of the following parameterized curves. 24. r(t)=3sint,sint,2costAlternative curvature formula Use the alternative curvature formula = |v a|/|v|3 to find the curvature of the following parameterized curves. 25. r(t) = 4 cos t,sin t, 2 cos tAlternative curvature formula Use the alternative curvature formula = |v a|/|v|3 to find the curvature of the following parameterized curves. 26. r(t) = et cos t, et sin t, et27E28E29E30E31E32E33E34EComponents of the acceleration Consider the following trajectories of moving objects. Find the tangential and normal components of the acceleration. 35. r(t) = t, 1 + 4t, 2 6tComponents of the acceleration Consider the following trajectories of moving objects. Find the tangential and normal components of the acceleration. 36. r(t) = 10 cos t, 10 sin tComponents of the acceleration Consider the following trajectories of moving objects. Find the tangential and normal components of the acceleration. 37. r(t) = et cos t, et sin t, etComponents of the acceleration Consider the following trajectories of moving objects. Find the tangential and normal components of the acceleration. 38. r(t) = t, t2 + 139E40EComputing the binormal vector and torsion In Exercises 2730, the unit tangent vector T and the principal unit normal vector N were computed for the following parameterized curves. Use the definitions to compute their unit binormal vector and torsion. 41. r(t) = 2 sin t, 2 cos tComputing the binormal vector and torsion In Exercises 2730, the unit tangent vector T and the principal unit normal vector N were computed for the following parameterized curves. Use the definitions to compute their unit binormal vector and torsion. 42. r(t) = 4 sin t, 4 cos t, 10t43E44E45EComputing the binormal vector and torsion Use the definitions to compute the unit binormal vector and torsion of the following curves. 46. r(t) = t, cosh t, sinh tComputing the binormal vector and torsion Use the definitions to compute the unit binormal vector and torsion of the following curves. 47. r(t) = 12t, 5 cos t, 5 sin t48EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The position, unit tangent, and principal unit normal vectors (r, T, and N) at a point lie in the same plane. b. The vectors T and N at a point depend on the orientation of a curve. c. The curvature at a point depends on the orientation of a curve. d. An object with unit speed (|v| = 1) on a circle of radius R has an acceleration of a = N/R. e. If the speedometer of a car reads a constant 60 mi/hr, the car is not accelerating. f. A curve in the xy-plane that is concave up at all points has positive torsion. g. A curve with large curvature also has large torsion.Special formula: Curvature for y = f(x) Assume that f is twice differentiable. Prove that the curve y = f(x) has curvature (x)=f(x)(1+f(x)2)3/2. (Hint: Use the parametric description x = t, y = f(t).)Curvature for y = f(x) Use the result of Exercise 50 to find the curvature function of the following curves. 51. f(x) = x252E53ECurvature for y = f(x) Use the result of Exercise 50 to find the curvature function of the following curves. 54. f(x) = ln cos x55ECurvature for plane curves Use the result of Exercise 55 to find the curvature function of the following curves. 56. r(t) = a sin t, a cos t (circle)Curvature for plane curves Use the result of Exercise 55 to find the curvature function of the following curves. 57. r(t) = a sin t, b cos t (ellipse)Curvature for plane curves Use the result of Exercise 55 to find the curvature function of the following curves. 58. r(t) = a cos3 t, a sin3 t) (astroid)Curvature for plane curves Use the result of Exercise 55 to find the curvature function of the following curves. 59. r(t) = t, at2) (parabola)Same paths, different velocity The position functions of objects A and B describe different motion along the same path, for t 0. a. Sketch the path followed by both A and B. b. Find the velocity and acceleration of A and B and discuss the differences. c. Express the acceleration of A and B in terms of the tangential and normal components and discuss the differences. 60. A: r(t) = 1 + 2t, 2 3t, 4t), B: r(t) = 1 + 6t, 2 9t, 12tSame paths, different velocity The position functions of objects A and B describe different motion along the same path, for t 0. a. Sketch the path followed by both A and B. b. Find the velocity and acceleration of A and B and discuss the differences. c. Express the acceleration of A and B in terms of the tangential and normal components and discuss the differences. 61. A: r(t) = t, 2t, 3t, B: r(t) = t2, 2t2, 3t2Same paths, different velocity The position functions of objects A and B describe different motion along the same path, for t 0. a. Sketch the path followed by both A and B. b. Find the velocity and acceleration of A and B and discuss the differences. c. Express the acceleration of A and B in terms of the tangential and normal components and discuss the differences. 62. A: r(t) = cos t, sin t, B: r(t) = cos 3t, sin 3tSame paths, different velocity The position functions of objects A and B describe different motion along the same path, for t 0. a. Sketch the path followed by both A and B. b. Find the velocity and acceleration of A and B and discuss the differences. c. Express the acceleration of A and B in terms of the tangential and normal components and discuss the differences. 63. A: r(t) = cos t, sin t, B: r(t) = cos t2, sin t2Graphs of the curvature Consider the following curves. a. Graph the curve. b. Compute the curvature. c. Graph the curvature as a function of the parameter. d. Identify the points (if any) at which the curve has a maximum or minimum curvature. e. Verify that the graph of the curvature is consistent with the graph of the curve. 64. r(t) = t, t2, for 2 t 2 (parabola)Graphs of the curvature Consider the following curves. a. Graph the curve. b. Compute the curvature. c. Graph the curvature as a function of the parameter. d. Identify the points (if any) at which the curve has a maximum or minimum curvature. e. Verify that the graph of the curvature is consistent with the graph of the curve. 65. r(t) = t sin t, 1 cos t), for 0 t 2 (cycloid)Graphs of the curvature Consider the following curves. a. Graph the curve. b. Compute the curvature. c. Graph the curvature as a function of the parameter. d. Identify the points (if any) at which the curve has a maximum or minimum curvature. e. Verify that the graph of the curvature is consistent with the graph of the curve. 66. r(t) = t, sin t, for 0 t (sine curve)Graphs of the curvature Consider the following curves. a. Graph the curve. b. Compute the curvature. c. Graph the curvature as a function of the parameter. d. Identify the points (if any) at which the curve has a maximum or minimum curvature. e. Verify that the graph of the curvature is consistent with the graph of the curve. 67. r(t) = t2/2, t3/3, for t 0Curvature of ln x Find the curvature of f(x) = ln x, for x 0, and find the point at which it is a maximum. What is the value of the maximum curvature?Curvature of ex Find the curvature of f(x) = ex and find the point at which it is a maximum. What is the value of the maximum curvature?70EFinding radii of curvature Find the radius of curvature (see Exercise 70) of the following curves at the given point. Then write an equation of the circle of curvature at the point. 71. r(t) = t, t2 (parabola) at t = 0Finding radii of curvature Find the radius of curvature (see Exercise 70) of the following curves at the given point. Then write an equation of the circle of curvature at the point. 72. y = ln x at x = 1Finding radii of curvature Find the radius of curvature (see Exercise 70) of the following curves at the given point. Then write an equation of the circle of curvature at the point. 73. r(t) = t sin t, 1 cos t (cycloid) at t =Designing a highway curve The function
r(t) =
whose graph is called a clothoid or Euler spiral (Figure A), has applications in the design of railroad tracks, roller coasters, and highways.
A car moves from left to right on a straight highway, approaching a curve at the origin (Figure B). Sudden changes in curvature at the start of the curve may cause the driver to jerk the steering wheel. Suppose the curve starting at the origin is a segment of a circle of radius a. Explain why there is a sudden change in the curvature of the road at the origin. (Hint: See Exercise 70.)
A better approach is to use a segment of a clothoid as an easement curve, in between the straight highway and a circle, to avoid sudden changes in curvature (Figure C). Assume the easement curve corresponds to the clothoid r(t), fir 0 ≤ t ≤ 1.2. Find the curvature of the easement curve as a function of t, and explain why this curve eliminates the sudden change in curvature at the origin.
Find the radius of a circle connected to the easement curve at point A (that corresponds to t = 1.2 on the curve r(t) so that the curvature of the circle matches the curvature of the easement curve at point A.
70. Circle and radius of curvature Choose a point P on a smooth curve C in the plane. The circle of curvature (or osculating circle) at P is the circle that (a) is tangent to C at P, (b) has the same curvature as C at P, and (c) lies on the same side of C as the principal unit normal N (see figure). The radius of curvature is the radius of the circle of curvature. Show that the radius of curvature is 1/κ, where κ is the curvature of C at P.
Curvature of the sine curve The function f(x) = sin nx, where n is a positive real number, has a local maximum at x = /(2n). Compute the curvature of f at this point. How does vary (if at all) as n varies?Parabolic trajectory In Example 7 it was shown that for the parabolic trajectory r(t) = t, t2, a = 0, 2 and a=21+4t2(N+2tT). Show that the second expression for a reduces to the first expression.Parabolic trajectory Consider the parabolic trajectory x=(V0cos)t,y=(V0sin)t12gt2, where V0 is the initial speed, is the angle of launch, and g is the acceleration due to gravity. Consider all times [0, T] for which y 0. a. Find and graph the speed, for 0 t T. b. Find and graph the curvature, for 0 t T. c. At what times (if any) do the speed and curvature have maximum and minimum values?78EZero curvature Prove that the curve r(t)=a+btp,c+dtp,e+ftp, where a, b, c, d, e, and f are real numbers and p is a positive integer, has zero curvature. Give an explanation.80EMaximum curvature Consider the superparabolas fn(x) = x2n, where n is a positive integer. a. Find the curvature function of fn, for n = 1, 2, and 3. b. Plot fn and their curvature functions, for n = 1, 2, and 3, and check for consistency. c. At what points does the maximum curvature occur, for n = 1, 2, 3? d. Let the maximum curvature for fn occur at x = zn. Using either analytical methods or a calculator, determine limnzn. Interpret your result.Alternative derivation of the curvature Derive the computational formula for curvature using the following steps. a. Use the tangential and normal components of the acceleration to show that v a = |v|3B. (Note that T T = 0.) b. Solve the equation in part (a) for and conclude that =vav3, as shown in the text.Computational formula for B Use the result of part (a) of Exercise 82 and the formula for to show that B=vava. 82. Alternative derivation of the curvature Derive the computational formula for curvature using the following steps. a. Use the tangential and normal components of the acceleration to show that v a = |v|3B. (Note that T T = 0.) b. Solve the equation in part (a) for and conclude that =vav3, as shown in the text.84EDescartes four-circle solution Consider the four mutually tangent circles shown in the figure that have radii a, b, c, and d, and curvatures A = 1/a, B = 1/b, C = 1/c, and D = 1/d. Prove Descartes result (1643) that (A+B+C+D)2=2(A2+B2+C2+D2).1RESets of points Describe the set of points satisfying the equations x2 + z2 = 1 and y = 2.
Graphing curves Sketch the curves described by the following functions, indicating the orientation of the curve. Use analysis and describe the shape of the curve before using a graphing utility.
r(t) = (2t + 1) i + tj
4RECurves in space Sketch the curves described by the following functions, indicating the orientation of the curve. Use analysis and describe the shape of the curve before using a graphing utility. 39. r(t) = 4 cos t i + j + 4 sin t k, for 0 t 2Curves in space Sketch the curves described by the following functions, indicating the orientation of the curve. Use analysis and describe the shape of the curve before using a graphing utility. 40. r(t) = eti + 2etj + k, for t 0Intersection curve A sphere S and a plane P intersect along the curve r(t) = sin t i + cos t j + sin t k, for 0 ≤ t ≤ 2π. Find equations for S and P and describe the curve r.
Vector-valued functions Find a function r(t) that describes each of the following curves. 8. The line segment from segment from P(2, 3, 0) to Q(1, 4, 9)Vector-valued functions Find a function r(t) that describes each of the following curves.
The line passing through the point P(4, −2, 3) that is orthogonal to the lines R(t) = and S(t) =
Vector-valued functions Find a function r(t) that describes each of the following curves. 10. A circle of radius 3 centered at (2, 1, 0) that lies in the plane y = 1Vector-valued functions Find a function r(t) that describes each of the following curves. 11. An ellipse in the plane x = 2 satisfying the equation y29+z216=1Vector-valued functions Find a function r(t) that describes each of the following curves. 12. The projection of the curve onto the xy -plane is the parabola y = x2, and the projection of the curve onto the xz-plane is the line z = x.13REIntersection curve Find the curve r(t) where the following surfaces intersect. 14. z = x2 5y2; z = 10x2 + 4y2 36Intersection curve Find the curve r(t) where the following surfaces intersect. 15. x2 + 7y2 + 2z2 = 9; z = y16RE17RE18RE19RE20RE21RE22RE23RE24REFinding r from r Find the function r that satisfies the given conditions. 25. r(t)=1,sin2t,sec2t; r(0)=2,2,2Finding r from r Find the function r that satisfies the given conditions. 26. r(t) = et,2e2t,6e3t ; r(0) = 1,3,127RE28RE29REVelocity and acceleration from position consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 30. r(t) = 53t3+1,t2+10t, for t 0Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. 31. r(t) = e4t+1,e4t,12e4t+1, for t 0Solving equations of motion Given an acceleration vector, initial velocity u0,v0, and initial position x0,y0, find the velocity and position vectors for t 0. 32. a(t)=1,4,u0,v0=4,3,x0,y0=0,233REOrthogonal r and r Find all points on the ellipse r(t) = 1, 8 sin t, cos t, for 0 t 2, at which r(t) and r(t) are orthogonal. Sketch the curve and the tangent vectors to verify your conclusion.Modeling motion Consider the motion of the following objects. Assume the x-axis is horizontal, the positive y-axis is vertical, the ground is horizontal, and only the gravitational force acts on an object.
Find the velocity and position vectors, for t ≥ 0.
Determine the time of flight and range of the object.
Determine the maximum height of the object.
A baseball has an initial position ft when it is hit at an angle of 60o with an initial speed of 80 ft/s.
36RE37REFiring angles A projectile is fired over horizontal ground from the ground with an initial speed of 40 m/s. what firing angles produce a range of 100m?39REBaseball motion A toddler on level ground throws a baseball into the air at an angle of 30 with the ground from a height of 2 ft. If the ball lands 10 ft from the child, determine the initial speed of the ball.41RE42RE43RE44REArc length Find the arc length of the following curves. 45. r(t)=sint,t+cost,4t, for 0t246REVelocity and trajectory length The acceleration of a wayward firework is given bya(t)=2j+2tk, for 0 t 3. Suppose the initial velocity of the firework is v(0) = i. a. Find the velocity of the firework, for 0 t 3. b. Find the length of the trajectory of the firework over the interval 0 t 3.48REArc length parameterization Find the description of the following curves that uses arc length as a parameter. 57. r(t)=t2,423t3/2,2t, for t 0