12.3 Velocity and Acceleration
750
c.
____ 2. The position vector describes the path of an object moving in space.
Find the speed of the object.
a.
b.
c.
d.
e.
____ 3. The position vector describes the path of an object moving in space. Find the acceleration
of the object.
a.
b.
c.
d.
e.
751 Chapter 12: Vector-Valued Functions
____ 4. The position vector describes the path of an object moving in space. Find the speed of
the object.
21
3 7
93
149
85
____ 5. The position vector describes the path of an object moving in space. Find the velocity
of the object.
a.
b.
c.
d.
e.
____ 6. The position vector describes the path of an object moving in
space. Find the velocity of the object.
a.
b.
c.
d.
e.
12.3 Velocity and Acceleration
752
____
7.
Use the given acceleration function and initial conditions to find the position at time t
= 1.
a.
b.
c.
d.
e.
____
8.
Use the given acceleration function and initial conditions to find the position at time t
= 1.
a.
b.
c.
d.
e.
____
9.
Use the given acceleration function and initial conditions to find the position at time t
= 3.
a.
b.
c.
d.
e.
____ 10. Determine the maximum height of a projectile fired at a height of 6 feet above the
ground with an initial velocity of 700 feet per second and at an angle of above the horizontal.
Use the model for projectile motion, assuming there is no air resistance. Round your answer to
three decimal places.
14,289.757 feet
14,292.757 feet
7,146.378 feet
7,149.378 feet
14,298.757 feet
753 Chapter 12: Vector-Valued Functions
____ 11. Determine the range of a projectile fired at a height of 5 feet above the ground with
an initial velocity of 1000 feet per second and at an angle of above the horizontal. Use the model
for projectile motion, assuming there is no air resistance. Round your answer to three decimal places.
15,627.500 feet
8,137.843 feet
31,252.187 feet
31,254.999 feet
16,268.138 feet
____ 12. A baseball, hit 9 feet above the ground, leaves the bat at an angle of and is caught
by an outfielder 9 feet above the ground and 500 feet from home plate. What is the initial speed
of the ball? Round your answer to two decimal places.
31.62 ft/sec
126.49 ft/sec
189.74 ft/sec
158.11 ft/sec
63.25 ft/sec
____ 13. A baseball, hit 5 feet above the ground, leaves the bat at an angle of and its caught by an
outfielder 5 feet above the ground and 500 feet from home plate. How high does the ball rise? Round
your answer to the nearest integer.
77 feet
10 feet
167 feet
118 feet
23 feet
____ 14. A baseball player at second base throws a ball 87 feet to the player at first base. The ball is
thrown 5 feet above the ground with an initial velocity of 50 miles per hour and at an angle of
above the horizontal. At which height does the player at first base catch the ball? Round
your answer to three decimal places.
7.576 feet
6.676 feet
5.476 feet
6.076 feet
5.576 feet
____ 15. The quarterback of a football team releases a pass at a height of 8 feet above the playing
field, and the football is caught by a receiver 38 yards directly downfield at a height of 4 feet.
The pass is released at an angle of with the horizontal. Find the speed of the football when it is
released. Round your answer to three decimal places.
33.169 feet per second
59.366 feet per second
19.462 feet per second
118.732 feet per second
190.165 feet per second
12.3 Velocity and Acceleration
754
____ 16.
The quarterback of a football team releases a pass at a height of 6 feet above the
playing field, and the football is caught by a receiver 42 yards directly downfield at a height of 3 feet.
The pass is released at an angle of
with the horizontal. Find the maximum height of the football.
Round your answer to one decimal place.
–295.8 feet
–194.1 feet
–278.9 feet
–224.3 feet
–197.7 feet
____ 17. The quarterback of a football team releases a pass at a height of 6 feet above the playing
field, and the football is caught by a receiver 34 yards directly downfield at a height of 3 feet.
The pass is released at an angle of with the horizontal. Find the time in seconds the receiver has
to reach the proper position after the quarterback releases the football. Round your answer to one
decimal place.
0.8 second
4.6 seconds
7.8 seconds
1.3 seconds
2.6 seconds
755 Chapter 12: Vector-Valued Functions
12.3 Velocity and Acceleration
Answer Section
12.4 Tangent Vectors and Normal Vectors
756
12.4 Tangent Vectors and Normal Vectors
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the unit tangent vector to the curve given below at the specified point.
a.
b.
c.
d.
e.
____ 2. Find the unit tangent vector to the curve given below at the specified point.
a.
b.
c.
d.
e.
757
Chapter 12: Vector-Valued Functions
____
3.
Find the unit tangent vector
for the line tangent to the space curve
at point
.
a.
b.
c.
d.
e.
____ 4. Find the principle unit normal vector to the curve given below at the specified point.
a.
b.
c.
d.
e.
12.4 Tangent Vectors and Normal Vectors
758
____ 5.
Find the principle unit normal vector to the curve given below at the specified point.
a.
b.
c.
d.
e.
____ 6. Find the principle unit normal vector to the curve given below at the specified point.
a.
b.
c.
d.
e.
759
Chapter 12: Vector-Valued Functions
____
7.
Find
at time
for the plane curve
. Round your answer to
three decimal places.
11.077
7.310
11.797
5.898
5.538
____
8.
Find
at time
for the plane curve
. Round your answer to
three decimal places.
a.
6.581
b.
4.818
c.
3.407
d.
2.154
e.
4.962
____
9.
Find
at time
for the plane curve
. Round your answer to
three decimal places.
–40.625
–39.209
–111.667
–46.016
–43.988
____ 10.
Find
at time
for the plane curve
. Round your answer to
three decimal places.
18.724
23.792
22.264
15.179
22.764
____ 11.
Find
at time
for the space curve
. Round your
answer to three decimal places.
a.
0.062
b.
0.071
c.
0.659
d.
0.000
e.
0.753
12.4 Tangent Vectors and Normal Vectors
760
____ 12.
Find
at time
for the space curve
.
16
0
5
25
4
____ 13. Find at time for the space curve . Round your answer
to three decimal places.
0.781
0.724
0.343
0.686
0.649
____ 14. Find at time for the space curve . Round your answer
to three decimal places.
0.970
0.465
0.930
1.814
1.060
____ 15.
Because of a storm, ground controllers instruct the pilot of a plane flying at an
altitude of 1.6 miles to make a
turn and climb to an altitude of 1.8 miles. The model for the path
of the plane during this maneuver is
where t is the
time in hours and is the distance in miles. Determine the speed of the plane. Round your answer to
the nearest integer.
50 mi/h
16 mi/h
54 mi/h
34 mi/h
79 mi/h
761 Chapter 12: Vector-Valued Functions
12.4 Tangent Vectors and Normal Vectors
Answer Section
12.5 Arc Length and Curvature
762
12.5 Arc Length and Curvature
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the length of the plane curve given below.
20
9
17
19
18
____ 2. A baseball is hit 5 feet above the ground at 140 feet per second and at an angle of with
respect to the ground. Find the vector-valued function for the path of the baseball.
a.
b.
c.
d.
e.
____ 3. A baseball is hit 3 feet above the ground at 80 feet per second and at an angle of with
respect to the ground. Find the maximum height. Round your answer to the nearest integer.
66 feet
65 feet
53 feet
56 feet
61 feet
____ 4. A baseball is hit 3 feet above the ground at 80 feet per second and at an angle of with
respect to the ground. Find the range. Round your answer to one decimal place.
203 feet
259.5 feet
275.7 feet
386.6 feet
763 Chapter 12: Vector-Valued Functions
e. 215.6 feet
____ 5. A baseball is hit 5 feet above the ground at 95 feet per second and at an angle of
with respect to the ground. Find the arc length of the trajectory. Round your answer to one
decimal place.
270.8 feet
298 feet
293.7 feet
239.1 feet
330.7 feet
____ 6.
Find the arc length for
over the interval
. Round your
answer to two decimal places.
a.
67.63
b.
73.37
c.
79.40
d.
49.75
e.
62.97
____ 7. Find the length of the space curve given below.
a.
b.
c.
d.
e.
____ 8. Find the curvature K of the curve given below.
a.
b.
c.
12.5 Arc Length and Curvature
764
d.
e.
____
9.
Find the curvature K of the curve
at the point
. Round your
answer to three decimal places.
a.
0.883
b.
0.994
c.
0.019
d.
0.012
e.
0.017
____
10.
Find the curvature K of the curve
at the point
.
Round your answer to three decimal places.
a.
7.286
b.
9.952
c.
0.021
d.
0.014
e.
0.009
____
11.
Find the curvature of the plane curve
at
. Round your answer to
three decimal places.
a.
0.498
b.
0.332
c.
0.015
d.
0.005
e.
0.003
____
12.
Find the radius of curvature of the plane curve
at
. Round
your answer to three decimal places.
a.
24.694
b.
85.333
c.
28.133
d.
34.510
e.
37.510
765 Chapter 12: Vector-Valued Functions
____
13.
Find the point on the curve
at which the curvature K is a
maximum.
a.
b.
c.
d.
e.
____
14.
Find the point on the curve given below at which the curvature K is zero.
a.
b.
c.
d.
e.
____ 15. The smaller the curvature in a bend of a road, the faster a car can travel. Assume that the
maximum speed around a turn is inversely proportional to the square root of the curvature. A car
moving on the path (x and y are measured in miles) can safely go 25 miles per hour at
. How fast can it go at ? Round your answer to two decimal places.
259.31 mi/h
149.71 mi/h
119.77 mi/h
75.30 mi/h
299.42 mi/h
12.5 Arc Length and Curvature
766
____ 16.
A 5700-pound vehicle is driven at a speed of 25 miles per hour on a circular
interchange of radius 90 feet. To keep the vehicle from skidding off course, what frictional force
must the road surface exert on the tires? Round your answer to one decimal place.
2,670.2 lb
3,337.8 lb
2,993.5 lb
2,660.9 lb
3,503.9 lb
767 Chapter 12: Vector-Valued Functions
12.5 Arc Length and Curvature
Answer Section