Deck 14: Vector-Valued Functions and Motion in Space

Question 1

A) x = 13t, y = t, z = t
B) x = 10 + t, y = t, z = t
C) x = 10 + 13t, y = 9 + t, z = 1
D) x = 10 + 13t, y = 9 + t, z = 0

Answer 1

C

Question 2

The position vector of a particle is r(t). Find the requested vector.

A) a(1) = -36i - j + 8k
B) a(1) = -9i + 8k
C) a(1) = 36i + 8k
D) a(1) = -36i + 8k

Answer 2

D

Question 3

The position vector of a particle is r(t). Find the requested vector.

Answer 3

C

Question 4

The position vector of a particle is r(t). Find the requested vector.

Question 5

The position vector of a particle is r(t). Find the requested vector.

Question 6

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.

Question 7

The position vector of a particle is r(t). Find the requested vector.

Question 8

The position vector of a particle is r(t). Find the requested vector.

Question 9

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors
at time t = 0.

Question 10

The position vector of a particle is r(t). Find the requested vector.

A) v(3) = 46i + 81j + 6k
B) v(3) = 25i - 27j - 3k
C) v(3) = 46i - 81j - 6k
D) v(3) = 38i - 81j - 6k

Question 11

The position vector of a particle is r(t). Find the requested vector.

Question 12

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.

Question 13

The position vector of a particle is r(t). Find the requested vector.

Question 14

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors
at time t = 0.

Question 15

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.
Find the acceleration vector. r(t) = (cos 4t)i + (5 sin t)j

A) a = (-4 cos 4t)i + (5 sin t)j
B) a = (-16 cos 4t)i + (-5 sin t)j
C) a = (-16 cos 4t)i + (-25 sin t)j
D) a = (16 cos 4t)i + (-5 sin t)j

Question 16

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.
Find the velocity vector. r(t) = (cot t)i + (csc t)j

Question 17

The position vector of a particle is r(t). Find the requested vector.

Question 18

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors
at time t = 0.

Question 19

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.
Find the acceleration vector. r(t) = (6 cos t)i + (9 sin t)j

A) a = (6 cos t)i + (9 sin t)j
B) a = (-6 cos t)i + (-9 sin t)j
C) a = (-6 sin t)i + (-9 cos t)j
D) a = (6 sin t)i + (9 cos t)j

Question 20

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors
at time t = 0.

Question 21

Evaluate the integral.

Question 22

Provide an appropriate response.

A) Path (4) and Path (2)
B) Path (1)
C) Path (2) and Path (3)
D) Path (3)

Question 23

Evaluate the integral.

Question 24

Solve the initial value problem.

Question 25

Evaluate the integral.

Question 26

Provide an appropriate response.
Show that if a particle's velocity vector is always orthogonal to its acceleration vector then the particle's speed is
constant.

Question 27

A) x = 10, y = -3t, z = -8 + t
B) x = 10t, y = -3, z = 1 - t
C) x = 10t, y = 3, z = 1 + t
D) x = 10t, y = -3, z = 1 - 8t

Question 28

Provide an appropriate response.

A) Path (2) and Path (4)
B) Path (4)
C) Path (3)
D) Path (1)

Question 29

Evaluate the integral.

Question 30

Evaluate the integral.

Question 31

Provide an appropriate response.

Question 32

Evaluate the integral.

Question 33

Evaluate the integral.

Question 34

Solve the initial value problem.

Question 35

Evaluate the integral.

Question 36

Provide an appropriate response.

Question 37

Solve the initial value problem.

Question 38

Solve the initial value problem.

Question 39

Provide an appropriate response.
The position of a particle is given by r(t) = sin 8t i + cos 3t j. Find the velocity vector for the particle.

A) cos 8t i - sin 3t j
B) 8 i + 3 j
C) 8 cos 8t i - 3 sin 3t j
D) 8 sin 8t i + 3 cos 3t j

Question 40

Question 41

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 2 ft
B) 74.4 ft
C) 227.8 ft
D) 6.8 ft

Question 42

Solve the problem.
At time t = 0 a particle is located at the point (4, -3, 2). It travels in a straight line to the point (6, -2, 1), has speed 2 at (4, -3, 2) and constant acceleration 6i - j - k. Find an equation for the position vector r(t) of the particle at time t.

Question 43

Solve the initial value problem.

Question 44

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 33 sec
B) 35 sec
C) 37 sec
D) It will never get that far downrange.

Question 45

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 26,749.4 m
B) 6687.3 m
C) 73.9 m
D) 65,536.0 m

Question 46

Solve the initial value problem.

Question 47

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 14.7 m/sec
B) 11.3 m/sec
C) 128.7 m/sec
D) 3.6 m/sec

Question 48

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 2.1 sec
B) 1.9 sec
C) 1.0 sec
D) 4.3 sec

Question 49

Solve the initial value problem.

Question 50

Provide an appropriate response.

Question 51

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface
Find the muzzle speed of a gun whose maximum range is 18.8 km. Round your answer to the nearest tenth.

A) 13.6 km/sec
B) 14.1 km/sec
C) 184.2 km/sec
D) 184.7 km/sec

Question 52

Provide an appropriate response.
Increasing the initial speed of a projectile by a factor of 6 increases its range by what factor? Assume the elevation is the same.

A) Factor of 6
B) Factor of 2.45
C) The elevation must be specified before an answer can be found.
D) Factor of 36

Question 53

Solve the initial value problem.

Question 54

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

Question 55

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

Question 56

Solve the initial value problem.

Question 57

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 71.0 ft
B) 84.0 ft
C) 82.0 ft
D) 78.8 ft

Question 58

Provide an appropriate response.
What two angles of elevation will enable a projectile to reach a target 17 km downrange on the same level as the gun if the projectile's initial speed is 420 m/sec? Assume there is no wind resistance.

Question 59

Provide an appropriate response.

Question 60

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface
An ideal projectile is launched from the origin at an angle of α radians to the horizontal and an initial speed of 175 ft/sec. Find the position function r(t) for this projectile.

Question 61

Calculate the arc length of the indicated portion of the curve r(t).

A) -1089
B) -99
C) 121
D) 1331

Question 62

Find the unit tangent vector of the given curve.

A) T = (5 sin 9t)i - (5 cos 9t)j
B) T = (sin 9t)i - (cos 9t)j
C) T = (135 sin 9t)i -(135 cos 9t)j
D) T = (5 cos 9t)i - (5 sin 9t)j

Question 63

Calculate the arc length of the indicated portion of the curve r(t).

A) -35
B) 45
C) -175
D) 225

Question 64

Provide an appropriate response.

A) Maximum height = 3.48 feet;
B) Maximum height = 0.21 feet; time to maximum height = 0.05 sec time to maximum height = -0.21 sec
C) Maximum height = 3.51 feet;
D) Maximum height = 8.71 feet; time to maximum height = 0.02 sec time to maximum height = 0.74 sec

Question 65

Calculate the arc length of the indicated portion of the curve r(t).

A) 9
B) 45
C) 27
D) 15

Question 66

Calculate the arc length of the indicated portion of the curve r(t).

A) 112
B) 56
C) 119
D) 238

Question 67

Calculate the arc length of the indicated portion of the curve r(t).

A) 270
B) 558
C) 450
D) 297

Question 68

Provide an appropriate response.

Question 69

Calculate the arc length of the indicated portion of the curve r(t).

Question 70

Provide an appropriate response.

Question 71

Find the unit tangent vector of the given curve.
r(t) = (7t cos t - 7 sin t)j + (7t sin t + 7 cos t)k

Question 72

Calculate the arc length of the indicated portion of the curve r(t).

A) 300
B) 4500
C) 18,000
D) 1200

Question 73

Provide an appropriate response.

Question 74

Provide an appropriate response.

A) Flight time = 3.97 sec; range = 424.4 feet
B) Flight time = 3.74 sec; range = 413.5 feet
C) Flight time = 2.82 sec; range = 393.5 feet
D) Flight time = 3.62 sec; range = 402.2 feet

Question 75

Find the unit tangent vector of the given curve.
r(t) = (10 - 2t)i + (2t - 3)j + (10 + t)k

Question 76

Find the unit tangent vector of the given curve.

Question 77

Calculate the arc length of the indicated portion of the curve r(t).

A) 1
B) 3
C) 0
D) 6

Question 78

Find the unit tangent vector of the given curve.

Question 79

Provide an appropriate response.

Question 80

Find the unit tangent vector of the given curve.