Deck 12: Vector-Valued Functions

Question 1

r(t) = 4t³ i + (17 + 2t)j is the position vector of a particle moving in a plane. Find the acceleration.

A) 12t² i + 2j
B) 12i
C) 24t i + 2j
D) 24t i
E) 24i

Answer 1

24t i

Question 2

An object in orbit has r_min = 10²⁴km and e = 0.49. Find r_max.

A) 2.92 *10²⁶km
B) 3.21 * 10²⁶km
C) 3.21 * 10²⁵km
D) 2.92 * 10²⁴km
E) 2.92 *10²⁵km

Answer 2

2.92 * 10²⁴km

Question 3

If, for an elliptical orbit, r_min = 10²⁵km and e = 0.55, find a, the semimajor axis.

A) 2.44*10²⁵km
B) 2.22 * 10²⁵km
C) 2.44 * 10²⁴km
D) 2.22 * 10²⁴km
E) 1.72 *10²⁵km

Answer 3

2.22 * 10²⁵km

Question 4

The position vector of a particle moving in a plane is given by . Find the velocity.

A)
B)
C)
D)
E)

Question 5

The velocity vector of a particle moving in a plane is given by . Find the acceleration vector.

A)
B)
C)
D)
E)

Question 6

An object in an elliptical orbit has and Find r_max.

A) km
B) km
C) km
D) km
E) km

Question 7

If an asteroid has an orbit with eccentricity 0.59 and semi-major axis a = 130000000, find its minimum distance from the center of the sun.

Question 8

If, for an elliptical orbit, r_max = 10²⁵km and e = 0.51, find a, the semimajor axis.

A) 3.25*10²⁴km
B) 6.62 *10²⁵km
C) 6.62 *10²⁴km
D) 5.96 * 10²⁴km
E) 7.28 * 10²⁴km

Question 9

A 2.03* 10³⁰-kg object is orbited by an object 1.42 *10¹⁵ m above its center. If the orbit is circular, find its velocity. G = 6.67 * 10^-11 m³/kg·s².

Question 10

The position vector of a particle moving in a plane is given by . Find the velocity.

A)
B)
C)
D)
E)

Question 11

Find the speed of a particle in a circular orbit with radius 10²¹m around an object of mass 10¹⁸kg. (G = 6.67 *10^-11m/kg·s²)

A) 1.50 * 10¹³m/s
B) 6.67*10^-14m/s
C) 3.87 * 10⁶m/s
D) 2.58 * 10^-7m/s
E) 4.91 *10^-7m/s

Question 12

If, for an elliptical orbit, r_min = 10²⁵km and e = 0.86, find a, the semimajor axis.

A) 7.14 * 10²⁴km
B) 7.14 *10²⁵km
C) 13.29 * 10²⁵km
D) 13.29 * 10²⁴km
E) 3.64 *10²⁵km

Question 13

Find the speed of a particle in a circular orbit with radius 10²²m around an object of mass 10²³kg. (G = 6.67 *10^-11m/kg·s²)

A) 1.50 *10⁹ m/s
B) 6.67 * 10^-10 m/s
C) 3.87 * 10⁴ m/s
D) 2.58 *10^-5 m/s
E) 4.91*10^-5m/s

Question 14

An object in an elliptical orbit has and Find r_min.

A) km
B) km
C) km
D) km
E) km

Question 15

Given G = 6.67 * 10^-11 m³/kg·s², find the radius of a circular orbit above a 10⁵⁰-kg mass, if the orbiting object has a speed of 51 m/s.

Question 16

If, for an elliptical orbit, r_max = 10²⁵km and e = 0.79, find a, the semi-major axis.

A) 1.17 *10²⁵km
B) 4.76 *10²⁵km
C) 5.59 * 10²⁵km
D) 5.59 * 10²⁴km
E) 1.17 * 10²⁴km

Question 17

If an asteroid has an orbit with eccentricity 0.59 and semi-major axis a = 130000000, find its maximum distance from the center of the sun.

Question 18

If an object orbits the sun with r_max = miles and r_min = miles, the elliptical orbit has eccentricity

A)
B)
C) 40
D)
E) 1

Question 19

If, for an elliptical orbit with semimajor axis a, r_min = a(1 - e) and r_max = a(1 + e), find the eccentricity if r_max = 1,100,000,000 km and r_min = 910,000,000 km.

Question 20

An object in orbit has r_max = 10²⁴km and e = 0.27. Find r_min.

A) 5.75 * 10²³km
B) 6.32 *10²³km
C) 5.46 *10²³km
D) 5.75 *10²⁶km
E) 6.32 * 10²⁶km

Question 21

Find the velocity, speed, and acceleration of a particle whose position is given by r(t) = e ^t i + e^2t j at t = 0 seconds, then sketch the path of the particle together with the velocity and acceleration vectors at t = 0 seconds.

Question 22

r(t) = t³ i - 2t j; 1 $\le$ t $\le$ 5. Find the displacement.

A) 124i - 8j
B) 125i - 8j
C) 124i + 10j
D) 124i + 8j
E) -124i - 8j

Question 23

Find the speed of a particle moving along the curve r(t) = (13 + t³)i + 4t j - t² k at t = 1.

A)
B)
C)
D)
E)

Question 24

If , find .

A)
B)
C)
D)
E)

Question 25

Find the velocity, speed, and acceleration of a particle whose position is given by r(t) = at seconds.

Question 26

A particle travels along a curve given by r(t) = 7 cos 3t i - 7 sin 3t j + 2k. Find the distance traveled by the particle during the time interval seconds.

Question 27

r(t) = 3t i + 6t j + 7t k is the position vector of a particle moving in a plane. Find the acceleration of the particle.

A) 3i + 6j + 9k
B) 18
C) 0
D) i + j + k
E) 3i - 6j - 9k

Question 28

r(t) = 31t + t³ i - 2t² j is the position vector of a particle moving in a plane. Find the acceleration at an arbitrary time t.

Question 29

r(t) = (-1 + t³)i - 2t j; 1 $\le$ t $\le$ 2. Find the distance.

A)
B)
C)
D) 9
E) 81

Question 30

If v = 2i and a = 5i - 7j, find a_T .

A) 5
B) 6
C)
D) 8
E) 9

Question 31

A shell is fired from a mortar at ground level with a velocity of 200 meters per second at an elevation of 60°. How far does the shell travel horizontally?

Question 32

r(t) = 4t³ i + (2t + 3)j is the position vector of a particle moving in a plane. Find the speed at t = 1.

A)
B) 12
C) 24
D) 0
E)

Question 33

Find the velocity, speed, and acceleration of a particle whose position is given by r(t) = 10 + e ^t i + e ^t cos t j + e ^t sin t k at t = 0 seconds.

Question 34

If v = 2i and a = 7i - 9j, find a_N .

A) 16
B) -16
C)
D) 9
E) -9

Question 35

Find the velocity of a particle moving along the curve r(t) = (-8 + t³)i + 4t j - t² k at t = 1.

A) 3i + 4j - 2k
B) 6i - 2j
C) 3i + 4j + 2k
D) 0
E) 3i - 4j - 2k

Question 36

A particle moves through 3-space in such a way that its velocity is . Find the coordinates of the particle at t = 1 second if the particle was initially at at t = 0 seconds.

Question 37

If , find when t = 1.

A)
B)
C)
D)
E)

Question 38

Find the acceleration of a particle moving along the curve r(t) = t³ i + (8 + 4t)j - t² k at t = 1.

A) 3i + 4j - 2k
B) 6i - 2k
C) 3i + 4j + 2k
D) 0
E) 3i - 4j - 2k

Question 39

Find the velocity, speed, and acceleration of a particle whose position is given by r(t) = $\langle$ 4 cos t, sin t $\rangle$ at seconds, then sketch the path of the particle together with the velocity and acceleration vectors at seconds.

Question 40

A particle travels along a curve given by r(t) = e ^t cos t i + e ^t sin t j + 12k. Find the distance traveled by the particle during the time interval 0 $\le$ t $\le$ $\pi$ seconds.

Question 41

Find the radius of curvature for x = t² + 4, y = t - 2, at t = 2.

Question 42

Find the curvature for r(t) = 17i + 2t j + 3t² k at t = 0.

A)
B)
C) 0
D) 1
E)

Question 43

If , find k(s).

A)
B)
C) 2
D) 4
E) 0

Question 44

At what points does 4x² + 25y² = 100 have minimum curvature?

A) (0, -4), (0, 4)
B) (-4, 0), (4, 0)
C) (-5, 0), (5, 0)
D) (0, -5), (0, 5)
E) (0, -2), (0, 2)

Question 45

At what point does have maximum curvature?

A) (0, -4)
B) (-1, 0)
C) (-5, 0)
D) (0, 0)
E) (0, 1)

Question 46

Find the curvature for .

A)
B)
C)
D) 6
E)

Question 47

If x = t³, y = 12 + t², then k(t) at t = 1 is

A)
B)
C) 0
D)
E)

Question 48

If y = cos x, find the curvature at .

A) 0
B) 1
C) -1
D)
E)

Question 49

Find the curvature for r(t) = (16 + sin t) i + (16 + cos t) j.

A) 1
B) -1
C) 0
D) sin²t - cos²t
E) - sin²t + cos²t

Question 50

If r(t) = (2t² - 5)i + (t - 2)j + (4t + 10)k, find the curvature k(t) at t = 1.

A)
B)
C)
D)
E)

Question 51

Find the radius of curvature for r(t) = $\langle$ 4 sin t, 2t - sin 2t - 3, cos 2t $\rangle$ at .

Question 52

Find the curvature for x = -11 + t³, y = 2t² at t = 1.

Question 53

Find the curvature for r(t) = e ^t i + e ^t j + 30k.

A) 0
B)
C)
D)
E) 2e

Question 54

Find the curvature for x = 2e ^t , y = 11 + 2e^-t at t = 0.

Question 55

Find the curvature for

A) 1
B)
C) 0
D) sin²t - cos²t
E) - sin²t + cos²t

Question 56

If and , then k(t) at t = 0 is

A)
B)
C)
D)
E)

Question 57

Find the curvature for

A)
B)
C)
D)
E)

Question 58

Find the curvature for r(t) = $\langle$ 6 cos 2t + 19, 6 sin 2t, 5t $\rangle$ at t = $\pi$ .

Question 59

If and , then k(t) at t = 0 is

A)
B)
C)
D)
E)

Question 60

Find the curvature for r(t) = cos t i + sin t j - 10k.

A)
B) 1
C) sin²t - cos²t
D) sin²t + cos²t
E) - sin²t + cos²t

Question 61

Let r(t) = 2t i + (10 + 3t)j. Find T(t).

A)
B)
C) 2i + 3j
D)
E)

Question 62

Let r(t) = e ^t i + e^2t j + (e^3t + 3)k. Find T(t) when t = 0.

A)
B) 6i - 6j + 2k
C)
D)
E) i

Question 63

Find the curvature for x = e ^t , y = 3 + e ^t cos t, z = e ^t sin t at t = 0.

Question 64

Sketch x = 2 cos t, y = 3 sin t for 0 $\le$ t $\le$ 2 $\pi$ . Calculate the radius of curvature at and sketch the oscillating circle.

Question 65

Let r(t) = (t² + 2)i + (10 + e ^t )j + e ^t k. Find B(t) for t = 0. Approximate your calculations to three decimal places.

A) 2i + 0.707j + 0.707k
B) 0.707j - 0.707k
C) 2i + 0.577j + 0.577k
D) 0.577j - 0.577k
E) 1.732i - 1.732j - 1.732k

Question 66

Find the curvature for x² + y² = 10x + 10 at (2, -4).

Question 67

Let r(t) = (t³ + 2)i + te ^t j + (9 + e ^t )k. Find T(t) for t = 0.

A)
B)
C)
D)
E)

Question 68

Let r(t) = (t² + 9)i + e ^t j + e ^t k. Find N(t) for t = 0. Approximate your calculations to three decimal places.

A) 0
B) i
C) 0.577j - 0.577k
D) i + 0.577j + 0.577k
E) 1.732i - 1.732j - 1.732k

Question 69

Let r(t) = (t² + 2)i + e ^t j + (12 + e ^t )k. Find N(t) for t = 0.

A)
B)
C)
D)
E)

Question 70

Find the curvature for at t = 0.

Question 71

Let r(t) = 4 i + (t + 2) j + 4 k. Find T(t) for t = 0.

A)
B)
C)
D)
E)

Question 72

Let r(t) = (-9 + t) i + t² j + t³ k. Find T(t) when t = 0.

A) i
B)
C)
D)
E)

Question 73

Let r(t) = (t² + 2)i + (e ^t + 4)j + e ^t k. Find N(t) for t = 0. Approximate your calculations to three decimal places.

A) 0
B) i
C) 0.577j + 0.577k
D) i + 0.577j + 0.577k
E) i - 1.732j - 1.732k

Question 74

Let r(t) = 3t² i + (4 + 4t²)j + 6t k. Find B(t).

A)
B) 0
C)
D)
E) i + j + k

Question 75

Sketch . Calculate the radius of curvature at x = 1 and sketch the oscillating circle.

Question 76

Let r(t) = (t³ + 2)i + te ^t j + (7 + e ^t )k. Find N(t) for t = 0.

A)
B)
C)
D)
E)

Question 77

Let r(t) = (t² + 2)i + e ^t j + (9 + e ^t )k. Find T(t) for t = 0.

A)
B)
C)
D)
E)

Question 78

Let r(t) = (t + 2)i + e ^t j + 12 k. Find T(t) for t = 0.

A)
B)
C)
D)
E)

Question 79

Let r(t) = e ^t i + (t + 2) j + 10 k. Find T(t) for t = 0.

A)
B)
C)
D)
E)

Question 80

Let r(t) = (t² + 2)i + e ^t j + ( + e ^t )k. Find B(t) for t = 0. Approximate your calculations to three decimal places.

A) 2i + 0.707j + 0.707k
B) 0.707j - 0.707k
C) 2i + 0.577j + 0.577k
D) 0.577j - 0.577k
E) 1.334i - 1.334j - 1.334k