2. Two-Dimensional Motion and Vectors Section 2
Vector Operations
• Use a traditional x-y coordinate system as shown below
on the right.
• The Pythagorean theorem and tangent function can be
used to add vectors.
– More accurate and less time-consuming than the graphical
method
4. Two-Dimensional Motion and Vectors Section 2
Vector Addition - Sample Problems
• 12 km east + 9 km east = ?
– Resultant: 21 km east
• 12 km east + 9 km west = ?
– Resultant: 3 km east
• 12 km east + 9 km south = ?
– Resultant: 15 km at 37° south of east
• 12 km east + 8 km north = ?
– Resultant: 14 km at 34° north of east
6. Two-Dimensional Motion and Vectors Section 2
Resolving Vectors into Components
• Opposite of vector addition
• Vectors are resolved into x and y components
• For the vector shown at right,
find the vector components vx
(velocity in the x direction) and
vy (velocity in the y direction).
Assume that that the angle is
20.0˚.
• Answers:
– vx = 89 km/h
– vy = 32 km/h
7. Two-Dimensional Motion and Vectors Section 2
Adding Non-Perpendicular Vectors
• Four steps
– Resolve each vector into x and y components
– Add the x components (xtotal = x1 + x2)
– Add the y components (ytotal = y1 + y2)
– Combine the x and y totals as perpendicular vectors
8. Two-Dimensional Motion and Vectors Section 2
Click below to watch the Visual Concept.
Visual Concept
Adding Vectors Algebraically
9. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Displacement
• Following the directions of the path, a pirate
walks 45.0 m north, then turns and walks 7.5 m
east. What single straight line displacement
could the pirate have taken to reach the
destination.
• Answer
– 4.5m at 9.5° E of N
10. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Displacement
• Emily passes a soccer ball 6.0 m directly across
the field to Kara, who then kicks the ball 14.5m
directly down the field to Luisa. What is the ball’s
total displacement as it travels between Emily
and Luisa?
• Answer
– 15.7m at 22° to the side of down the field
11. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Displacement
• A humming bird flies 1.2 m along a straight path
at a height of 3.4 m above the ground. Upon
spotting a flower below, the humming bird drops
directly downward 1.4 m to hover in front of the
flower. What is the hummingbird’s
displacement?
• Answer
– 1.8 m at 49° below the horizontal
12. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Resolving Vectors
• Find the horizontal and vertical component of
the 125 m displacement of a superhero who flies
down from the top of a tall building at an angle of
25 ° below the horizontal.
Answer
– 110 m
– -53 m
13. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Resolving Vectors
• Find the horizontal and vertical component of
the child’s toboggan ride down a hill if the angle
of the hill is 30.5 ° to the horizontal and the hill is
23m long.
Answer
– 19.8 m
– -11.7 m
14. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Resolving Vectors
• Find the horizontal and vertical component of
the truck’s velocity, if the truck drives up a hill
with a 15° incline at a constant speed of 22 m/s.
Answer
– 21m/s
– 5.7m/s
15. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Algebraically
• A football player runs directly down the field for
35 m before turning to the right at an angle of
25° from his original direction an running an
additional 15 m before getting tackled. What is
the total displacement of the player
Answer
– 49 m at 7.3° to the right of down field
16. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Algebraically
• A plane travels 25 km at an angle of 35 to the
ground, then changes direction and travels 515
km at an angle of 22° to the ground. What is the
displacement of the plane?
Answer
– 540,000 m at 22° above the horizontal
17. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Algebraically
• During a rodeo a clown runs 8.0 m north, turns
35 ° east of north, and runs 3.5 m. then , after
waiting for the bull to come near, the clown turns
due east and runs 5.0 m to exit the arena. What
is the clown’s displacement?
Answer
– 13.5 m at 37° north of east
18. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Resolving Vectors
• Find the magnitude resultant velocity.
A fish swimming a t 3.0 m/s across a river that
moves at 5.0 m/s?
What is the direction?
• Answer
– 5.8 m/s
– 59° downriver from its intended path
19. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Resolving Vectors
• Find the magnitude resultant velocity.
A surfer travelling at 1.0 m/s across a wave that
is traveling at 6.0 m/s?
What is the direction?
• Answer
– 6.1 m/s
– 9.5° from the direction the wave is traveling
20. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Resolving Vectors
• Find the component vectors along the directions
noted in parentheses.
A car displaced northeast by 10.0 km (north and
east
• Answer
– 7070 m, 7070 m
21. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Resolving Vectors
• Find the component vectors along the directions
noted in parentheses.
A duck accelerating away from a hunter at 2.0
m/s2 at an angle of 35° to the ground (horizontal
and vertical)
• Answer
– 1.6 m/s2, 1.1 m/s2
22. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Resolving Vectors
• Find the component vectors along the directions
noted in parentheses.
A submarine moving at 10.0 m/s toward the
surface at an angle of 35 ° to the ground
(horizontal and vertical)
• Answer
– 8.2 m/s, 5.7 m/s
23. Two-Dimensional Motion and Vectors Section 2
Classroom Practice Resolving Vectors
• Find the resultant displacement of a fox
searching for mice in a prairie. First the fox
heads 55° north of west for 10.0 m then it turns
and heads west for 5.0m.
• Answer
– 13.5 m at 37° north of east
Editor's Notes
Direction means north, south, east, west, up, or down. It does not mean increasing or decreasing. So even though the temperature may be going “up,” it is really just increasing and has no direction.
Remind students that the Pythagorean theorem can only be used with right triangles.
For the first two items, have students predict the answer before showing it. They generally have no trouble with these two problems. Point out that the process is the same if it is km/h or m/s2. Only the units change. These problems do not require trigonometry because the vectors are in the same direction (or opposite directions).
For the third problem, most students will probably remember the Pythagorean theorem and get the magnitude, but many will fail to get the direction or will just write southeast.
Show students how to use the trig identities to determine the angle. Then, explain why it is south of east and not east of south by showing what each direction would look like on an x-y axis. If they draw the 9 km south first and then add the 12 km east, they will get an answer of 53° east of south (which is the same direction as 37° south of east). After your demonstration, have students solve the fourth problem on their own, and then check their answers. Review the solution to this problem also.
Insist that students place arrows on every vector drawn. When they just draw lines, they often draw the resultant in the wrong direction.
You might find the PHet web site helpful (http://phet-web.colorado.edu/web-pages/index.html). If you go to the Math simulations, you will find a Vector Addition (flash version). You can download these simulations so your access to the internet is not an issue. You can show both the resultant and components using this simulation. Students could also use this at home to check their solutions to problems.
Review these trigonometry definitions with students to prepare for the next slide (resolving vectors into components).
Review the first solution with students, and then let them solve for the second component.
Explain the four steps using the diagram.
Show students that d1 can be resolved into x1 and y1 . Similarly for d2. Then, the resultant of d1 and d2 (dashed line labeled d) is the same as the resultant of the 4 components.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow students some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Student should note that the 4.5 km south is subtracted from the y component of the first vector that is directed north.
