Calculus 2nd Edition Rogawski

770 C H A P T E R 14 CALCULUS OF VECTOR-VALUED FUNCTIONS
In Exercises 30–33, use Eq. (3) to find the coefficients a T and a N as a
function of t (or at the specified value of t).
30. r(t) = 〈 t 2 ,t 3〉 31. r(t) = 〈 t,cos t,sin t 〉
32. r(t) = 〈 t −1 , ln t,t 2〉 , t = 1
47. A space shuttle orbits the earth at an altitude 400 km above the
earth’s surface, with constant speed v = 28,000 km/h. Find the magnitude
of the shuttle’s acceleration (in km/h 2 ), assuming that the radius
of the earth is 6378 km (Figure 12).
33. r(t) = 〈 e 2t ,t,e −t 〉 , t = 0
In Exercise 34–41, find the decomposition of a(t) into tangential and
normal components atthe point indicated, as in Example 6.
34. r(t) = 〈 e t , 1 − t 〉 , t = 0
〈
35. r(t) = 13
t 3 , 1 − 3t〉
, t = −1
〈
36. r(t) = t, 1 2 t2 , 1 6 t3〉 , t = 1
〈
37. r(t) = t, 1 2 t2 , 1 6 t3〉 , t = 4
38. r(t) = 〈 4 − t,t + 1,t 2〉 , t = 2
39. r(t) = 〈 t,e t ,te t 〉 , t = 0
40. r(θ) = ⟨cos θ, sin θ, θ⟩, θ = 0
41. r(t) = ⟨t,cos t,t sin t⟩, t = π 2
42. Let r(t) = 〈 t 2 , 4t − 3 〉 . Find T(t) and N(t), and show that the decomposition
of a(t) into tangential and normal components is
( ) ( )
2t
4
a(t) = √ T + √ N
t 2 + 4 t 2 + 4
43. Find the components a T and a N of the acceleration vector of a particle
moving along a circular path of radius R = 100 cm with constant
velocity v 0 = 5 cm/s.
44. In the notation of Example 5, find the acceleration vector for a
person seated in a car at (a) the highest point of the Ferris wheel and
(b) the two points level with the center of the wheel.
45. Suppose that the Ferris wheel in Example 5 is rotating clockwise
and that the point P at angle 45 ◦ has acceleration vector a = ⟨0, −50⟩
m/min 2 pointing down, as in Figure 11. Determine the speed and tangential
acceleration of the Ferris wheel.
y
FIGURE 12 Space shuttle orbit.
48. A car proceeds along a circular path of radius R = 300 m centered
at the origin. Starting at rest, its speed increases at a rate of t m/s 2 . Find
the acceleration vector a at time t = 3 s and determine its decomposition
into normal and tangential components.
49. A runner runs along the helix r(t) = ⟨cos t,sin t,t⟩. When he is
at position r ( π
2
)
, his speed is 3 m/s and he is accelerating at a rate of
1
2 m/s2 . Find his acceleration vector a at this moment. Note: The runner’s
acceleration vector does not coincide with the acceleration vector
of r(t).
50. Explain why the vector w in Figure 13 cannot be the acceleration
vector of a particle moving along the circle. Hint: Consider
the sign of w · N.
w
N
FIGURE 13
51. Figure 14 shows acceleration vectors of a particle moving
clockwise around a circle. In each case, state whether the particle is
speeding up, slowing down, or momentarily at constant speed. Explain.
Ferris wheel
45°
x
(A) (B) (C)
FIGURE 14
FIGURE 11
46. At time t 0 , a moving particle has velocity vector v = 2i and acceleration
vector a = 3i + 18k. Determine the curvature κ(t 0 ) of the
particle’s path at time t 0 .
52. Prove that a N =
∥a × v∥
.
∥v∥
53. Suppose that r = r(t) lies on a sphere of radius R for all t. Let
J = r × r ′ . Show that r ′ = (J × r)/∥r∥ 2 . Hint: Observe that r and r ′
are perpendicular.