Vectors
Chapter 3
Chapter Objectives
The student will be able to:
1.
2.
3.
4.
5.
6.
7.
Distinguish between a scalar and a vector.
Determine the resultant vector by graphically adding all the individual vectors.
Determine the resultant vector of two perpendicular vectors using trigonometry.
Determine the two perpendicular vector components of a resultant vector using
trigonometry.
Solve problems involving relative velocity.
Apply the independence of vectors for motion.
Determine the resultant vector of any two or more vectors, regardless of the angle
between the vectors using trigonometry.
ALCOS :
1.
Explain linear, uniform circular, and projectile motions using one and two dimensional
vectors.
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You must draw the vector diagrams, even if the vectors are
draw in this folder – you must copy them into your homework!
Vectors have magnitude and DIRECTION!
Simple Vectors
Draw the Vectors!
1.
While following the directions on a treasure map, a pirate walks 45.0 m north and then
turns and walks 7.50 m east. What is the pirate’s displacement?
2.
A hummingbird, 3.40 m above the ground, flies 1.20 m along a straight path. Upon
spotting a red flower below, the hummingbird drops directly downward 1.40 m to hover
in front of the flower. What is the hummingbird’s total displacement?
3.
You hike 1.35 km West then turn North and hike another 1.75 km. What is you
displacement?
4.
Kangaroos can easily jump as far as 8.00 m. If a kangaroo makes five such jumps
westward, then turns north and makes seven more jumps. Find the kangaroo's
displacement.
5.
Woodcocks are the slowest birds: their average speed during courtship displays can be
as low at 8.00 km/hr. Suppose a woodcock flies east for 15.0 min. If then it turns and
flies north for 22.0 min. Calculate the woodcock's displacement.
6.
An ostrich cannot fly, but it is able to run fast. Suppose an ostrich runs west for
7.95 sec and then runs 161 m south, so that the magnitude of the ostrich's resultant
displacement is 226 m. Calculate eastward velocity.
Vector Components
Draw the vectors!
In Virginia during 1994, Elmer Treet reached a speed of 372 km/hr on his motorcycle.
Suppose Treet rode at this speed for 8.70 sec in a direction of 60.0°.
7.
What was Treet's displacement east?
8.
What was Treet’s displacement north?
The straight stretch of Interstate Highway 5 from Mettler, California, to a point near
Buttonwillow, California, is 53.0 km long and makes an angle of 48.7° north of west.
9.
What is the northern component of this highway segment?
10. What is the western component of this highway segment?
The longest delivery flight ever made by a twin-engine commercial jet took place in 1990. The
plane covered a total distance of 14,890 km from Seattle, Washington to Nairobi, Kenya in
18.5 hours. The plane flew in a straight line between the two cities in the direction of 335°.
11. What is the eastward component of the average velocity?
12. What is the southward component of the average velocity?
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13.
A bowling ball is released at the near right corner of a bowling lane and travels 19.1 m at
an angle of 3.00° with respect to the lane’s length. The ball reaches the far left corner
of the lane, where it knocks over the “7” pin. What is the width of the lane?
14.
How fast must a truck travel to stay beneath an airplane that is moving 105 km/h at an
angle of 25.0° to the ground?
Relative Motion
Draw the Vectors!
An escalator is 20.0 m long. If a person stands on the escalator, it takes 50.0 s to ride to the
top.
15. If a person walks up the moving escalator with a speed of 0.500 m/s relative to the
escalator, how long does it take the person to get to the top?
16. If a person walks down the “up” escalator with a speed of 0.500 m/s relative to the
escalator. How long (in minutes) does it take to reach the bottom?
17.
You are piloting a small plane, and you want to reach an airport 450.0 km due south, in
3.00 hours. A wind is blowing from the west a 50.0 km/hr. What heading and airspeed
should you choose to reach your destination in time?
A ferryboat has a speed of 9.50 m/s in still water. The ferryboat intends to go north,
directly across a river, that has an easterly current of 3.75 m/s.
18. What must the ferryboat’s direction be from the shore?
19. What is the ferryboat’s velocity relative to the shore?
20. If the river is 1.25 km wide, how many minutes will it take the ferryboat to cross the
river?
The world’s fastest current is in Slingsby channel, Canada, where the velocity of the water
reaches 30.0 km/hr. Suppose a motorboat crosses the channel perpendicular to the bank at a
velocity of 18.0 km/hr relative to the shore. Assume the river is flowing East and the boat is
trying to head North across the channel.
21. What is the velocity of the motorboat relative to the water?
22. What direction must the motorboat head to go straight (perpendicular to the bank)
across the channel? Use the bank as your reference.
23. If the channel is 1.55 km wide, how long does it take the motorboat to cross the
channel?
Vectors at any Angle
Draw the Vectors!
24. A quarterback takes the ball from the line of scrimmage, runs west for 10.0 yards, and
then runs north for 15.0 yards. At this point, he throws a 50.0 yard forward pass
straight down the field (which is east). What is the football’s resultant displacement?
25. A plane travels 2.50 km at an angle of 35.0° to the ground and then changes direction
and travels 5.20 km at an angle of 22.0° to the ground. What is the plane’s total
displacement?
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26. A person walks the path shown
at the right. The total trip
consists of four straight-line
paths. At the end of the walk,
what is the person’s resultant
displacement measured from
the starting point? Please draw
the vectors in your homework.
27. Find the sum of these four vector forces: 12.0 N at 35.0º, 31.0 N at 125.0º, 8.40 N at
215.0º, and 24.0 N at 305.0º. (Hint: N stands for Newton, the SI unit of force. The
component method allows the addition of any vectors - forces as well as displacements
and velocities.)
28. The eye of a hurricane passes over
Grand Bahama Island. It is moving in a
direction 60.0° north of west with a
speed of 41.0 km/h. Exactly three hours
later, the course of the hurricane shifts
due north, and its speed slows to
25.0 km/h. How far from Grand Bahama
is the hurricane 4.50 h after it passes
over the island?
29. A map suggests that Atlanta is 730.0 miles in a direction of 5.00º north of east from
Dallas. The same map shows that Chicago is 560.0 miles in a direction of 21.0º west of
north from Atlanta. Modeling the Earth as flat, use this information to find the
displacement from Dallas to Chicago.
Vectors Review
Draw the Vectors!
An airplane is heading due North at a velocity of 300.0 km/hr. A wind begins to blow from the
West at a velocity of 55.0 km/hr.
30. Calculate the apparent velocity of the plane as viewed from the ground.
31. How far off course will the plane be after 30 minutes if the pilot takes no corrective
action?
32. Sally starts out hiking south and travels 4.50 km. When she comes to a canyon, she turns
east and travels 12.0 km before stopping for the night. How far did she travel this first
day and in what direction?
33. The next day, Sally (in the previous problem) consults her map and sets out on a path
35.0° south east, hiking 7.75 km before coming to a river. She turns south and follows
the river for 11.0 km, and again she stops for the night. How far did she travel the
second day, and in what direction?
34. A car travels due east with a speed of 50.0 km/hr. Rain is falling vertically with respect
to Earth. The traces of the rain on the side windows of the car make an angle of 60.0º
with the vertical. Find the velocity of the rain falling vertically to Earth.
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35. A sailboat travels south with a speed of 12.0 km/hr with respect to the water. Suppose
the boat encounters a current that has a velocity with respect to Earth of 4.00 km/hr
at 15.0° south of east. What is the sailboat’s resultant velocity with respect to Earth?
36. Towns A and B are 80.0 km apart. A friend (car 1) decides
to drive from to Town A and meet another friend (car 2) at
the Lake. The two friends leave at the same time and drive
for 2.50 hours in the directions shown. Car 1 has a velocity
of 90.0 km/hr. If the cars arrive at the same time at the
like, what is the velocity of car 2?
**OR PHYSICS CLASS!
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Answers for Vectors
1.
45.6 m, 80.5°
2.
1.84 m, 311°
3.
2.21 km, 128°
4.
68.6 m, 126°
5.
3.55 km, 55.7°
6.
20.0 m/s,180°
7.
450 m East
8.
779 m North
9.
35.0km West
10.
39.8 km North
11.
730 km/hr East
12.
340 km/hr South
13.
1.00 m
14.
95.2 km/hr
15.
22.2 sec
16.
3.33 min
17.
158 km/hr, 252°
18.
66.8° from the west bank of the shore (upstream)
19.
8.73 m/s
20.
2.39 min
21.
35.0 km/hr
22.
31.0° from the east bank of the shore (upstream)
23.
5.17 min
24.
42.7 yards, 20.6°
25.
7.66 km, 26.2°
26.
240 m, 237°
27.
7.87 N, 97.8°
28.
157 km, 113°
29.
788 miles, 48°
30.
305 km/hr, 79.6°
31.
27.5 km
32.
12.8 km, 339°
33.
16.7 km, 292°
34.
28.0 km/hr, straight down
35.
13.6 km/hr, 287°
36.
68.6 km/hr
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