1. Name: __________________________ Period: _____ Date: __________
Vectors
Pearland ISD Physics
Mrs. Akibola
Warm-up: Have students do “Introduction to Orienteering” as soon as class starts.
NOTES

3. A scalar quantity is one that can be described by a single number:
Scalars and Vectors
A scalar quantity is one that can be described by a single number:
Examples:
temperature, speed, mass
A vector quantity deals inherently with both magnitude and direction:
Acceleration, velocity, force, displacement

4. Identify the following as scalars or vectors – not in your notes 
The acceleration of an airplane as it takes off
The number of passengers and crew on the airplane
The duration of the flight
The displacement of the flight
The amount of fuel required for the flight
vector
scalar
scalar
vector
scalar

5. Name: __________________________ Period: _____ Date: __________
Representing vectors
Vectors can be represented graphically.
The direction of the arrow is the direction of the vector.
The length of the arrow tells the magnitude
Vectors can be moved parallel to themselves and still be the same vector.
Vectors only tell magnitude (amount) and direction, so a vector can starts anywhere
8 N
4 N
We can also represent vectors graphically. Follow directions under “Resolving vectors” in the lab.
(1)
Since vectors are just magnitude and direction, not where they start, we can move them around and they are still the same vector.
(2)
magnitude and direction, can we “add” them just by adding the numbers? What would happen if we just added all the numbers in our list of vectors in the lab?
NOTES

6. Name: __________________________ Period: _____ Date: __________
Adding vectors
The sum of two vectors is called the resultant.
To add vectors graphically, draw each vector to scale.
Place the tail of the second vector at the tip of the first vector. (tip-to-tail method)
Vectors can be added in any order.
To subtract a vector, add its opposite.
(1)
In the lab, you should have seen how to add vectors. First, what did you do? You moved one vector, and then, from the endpoint of that one, you moved to the next vector, and so on. How could we draw this? (Draw it out on board.) Notice that I am doing the displacements tip to tail. I start vector 2 where vector 1 ended, etc.
(2)
(3)
In your lab, did it matter what order you added vectors? Different route, but same ultimate ending point.
(4)
Now, subtraction is how we “undo” addition. If we add 3, we can subtract 3 to get back to original number.
How do we undo adding a vector?
(5)
NOTES

7. Name: __________________________ Period: _____ Date: __________
Adding Vectors + = + = = =
(1)
In the lab, you should have seen how to add vectors. First, what did you do? You moved one vector, and then, from the endpoint of that one, you moved to the next vector, and so on. How could we draw this? (Draw it out on board.) Notice that I am doing the displacements tip to tail. I start vector 2 where vector 1 ended, etc.
(2)
(3)
In your lab, did it matter what order you added vectors? Different route, but same ultimate ending point.
(4)
Now, subtraction is how we “undo” addition. If we add 3, we can subtract 3 to get back to original number.
How do we undo adding a vector?
(5) + =
NOTES

8. Vector Addition and Subtraction
Often it is necessary to add one vector to another.

9. Vector Addition and Subtraction
3 m
5 m
8 m

10. Adding Vector Problem 750 N downward
A parachutist jumps from a plane. He has not pulled is parachute yet. His weight or force is 800 N downward. The wind is applying a small drag force of 50 N upward. What is the vector sum of the forces acting on him?
750 N downward

11. Adding perpendicular vectors
Name: __________________________ Period: _____ Date: __________
Adding perpendicular vectors
Perpendicular vectors can be easily added and we use the Pythagorean theorem to find the magnitude of the resultant.
Use the tangent function to find the direction of the resultant.
What if we want to add vectors without having to draw them. Point out to students in the lab, question 3 asks how to find the magnitude of resultant vector if DELx = +6 and DEL Y = +8. These vectors are perpendicular, i.e. lie along x and y.
(1)
Using Pythagorean theorem, can find length of hypotenuse. Draw for students.
(2)
How do we get angle? Remember, tan angle = length of opposite side/length of adjacent
Find magnitude and direction of angle from lab. Then do Practice 3A, #2 as GP.
NOTES

12. Vector Addition and Subtraction

13. 1.6 Vector Addition and Subtraction
2.00 m
6.00 m

14. Vector Addition and Subtraction
2.00 m
6.00 m

15. Adding perpendicular Vectors – this slide is not in your notes
Note: in this example, In order to use the tip-to-tail Method, Vector B must be moved.

16. Vector Addition and Subtraction This slide is not in your notes
YOU MAY WANT TO MAKE A NOTE OF THIS!!
When a vector is multiplied
by -1, the magnitude of the
vector remains the same, but
the direction of the vector is
reversed.

17. Vector Addition and Subtraction

18. Name: __________________________ Period: _____ Date: __________
Multiplying Vectors
A vector multiplied or divided by a scalar results in a new vector.
Multiplying by a positive number changes the magnitude of the vector but not the direction.
Multiplying by a negative number changes the magnitude and reverses the direction.
What if I said, I was driving North at 30 mph. When I passed the cop, I tripled my speed. So the magnitude of the velocity changed, but not direction.
V became 3 V
(4)
NOTES

19. Resolving vectors into components.
Name: __________________________ Period: _____ Date: __________
Resolving vectors into components.
Any vector can be resolved, that is, broken up, into two vectors, one that lies on the x-axis and one on the y-axis.
Previously, we saw how to add perpendicular vectors, that is, along x and y axes. But what if we want to add vectors that don’t lie along x and y? What do we do?
Well, we can try and find vectors that do lie along x and y that can take the place of our vectors. Then we can add up those vectors to find our resultant. The first step then is to resolve, that is break up, our vectors into ones that lie along the x and y axis.
(Do above practice assuming angle is 20 degrees, and length is 6 m/s.)
Sometimes all we want to know is one component of a vector. For example, Problem 1 from your homework. (3B, p. 94)
NOTES

20. Useful formulas when calculating vectors
Write this on notebook paper if you need to and staple it to your notes

21. Inverse functions to calculate angles
Write this on notebook paper if you need to and staple it to your notes

22. 1.7 The Components of a Vector

23. 1.7 The Components of a Vector

24. Resolving Vectors - Practice Does 0.5 + 0.7 = 1 ?
Name: __________________________ Period: _____ Date: __________
Resolving Vectors - Practice Does = 1 ?
Consider two forces acting on an object, one force with 0.7 N of force at 600 above the horizontal and to the right, and the other pulling with 0.5 N of force at 450 above the horizontal to the left.
How much total force is produced by the two forces?
Show forces in equilibrium. Explain what the students are seeing.
Why doesn’t center move? (Forces balance out) How is this possible? (Resultant equals zero)
Let’s add these two vectors up and see if they indeed are the opposite of the bottom one.
NOTES

25. Adding non-perpendicular vectors
Name: __________________________ Period: _____ Date: __________
Adding non-perpendicular vectors
Resolve each vector into x and y components, using sin and cos.
Add the x components together to get the total x component. Add the y component together to get the total y component.
Find the magnitude of the resultant using Pythagorean theorem.
Find the direction of the resultant using the inverse tan function.
First, sketch situation.
Then, resolve vectors into components.
Once we have our vectors resolved in x and y components, adding them up is easy. We simply add up our components separately.
Next we find the magnitude of the resultant
NOTES

26. Adding non-perpendicular vectors
Name: __________________________ Period: _____ Date: __________
Adding non-perpendicular vectors
Resolving Vector 1:
F1 = 0.7 N
F1Y
First, sketch situation.
60o
F1X
NOTES

27. Adding non-perpendicular vectors
Name: __________________________ Period: _____ Date: __________
Adding non-perpendicular vectors
Resolving Vector 1:
F1 = 0.7 N
F1Y
First, sketch situation.
Then, resolve vectors into components.
60o
F1X
NOTES

28. Adding non-perpendicular vectors
Name: __________________________ Period: _____ Date: __________
Adding non-perpendicular vectors
Resolving Vector 1: X Y F1 .4 .6 F2 FR
First, sketch situation.
Then, resolve vectors into components.
NOTES

29. Adding non-perpendicular vectors
Name: __________________________ Period: _____ Date: __________
Adding non-perpendicular vectors
Resolving Vector 2:
F2 = 0.5 N
F2Y
45o
First, sketch situation.
F2X
NOTES

30. Adding non-perpendicular vectors
Name: __________________________ Period: _____ Date: __________
Adding non-perpendicular vectors
Resolving Vector 2
F2 = 0.5 N
F2Y
45o
First, sketch situation.
Then, resolve vectors into components.
F2X
NOTES

31. Adding non-perpendicular vectors
Name: __________________________ Period: _____ Date: __________
Adding non-perpendicular vectors
Resolving Vector 2: X Y F1 .4 .6 F2
-.4 FR
First, sketch situation.
Then, resolve vectors into components.
NOTES

32. Adding non-perpendicular vectors
Name: __________________________ Period: _____ Date: __________
Adding non-perpendicular vectors
Add the x components to get the resultant x component. X Y F1 .4 .6 F2
-.4 FR
Once we have our vectors resolved in x and y components, adding them up is easy. We simply add up our components separately.
NOTES

33. Adding non-perpendicular vectors
Name: __________________________ Period: _____ Date: __________
Adding non-perpendicular vectors
Add the y components to get the resultant y component. X Y F1 .4 .6 F2
-.4 FR
1.0
Once we have our vectors resolved in x and y components, adding them up is easy. We simply add up our components separately.
NOTES

34. Adding non-perpendicular vectors
Name: __________________________ Period: _____ Date: __________
Adding non-perpendicular vectors
Use the Pythagorean Theorem to find the magnitude of the resultant.
Use the inverse tan function to find the direction of the resultant. X Y F1 .4 .6 F2
-.4 FR
1.0
Once we have our vectors resolved in x and y components, adding them up is easy. We simply add up our components separately.
NOTES

35. Adding non-perpendicular vectors
Name: __________________________ Period: _____ Date: __________
Adding non-perpendicular vectors
Use the Pythagorean Theorem to find the magnitude of the resultant.
Use the inverse tan function to find the direction of the resultant.
Once we have our vectors resolved in x and y components, adding them up is easy. We simply add up our components separately. Next we find the magnitude of the resultant
(Do practice 3C #1 for GP.)
Do Prac. 3C 2 for IP.
What does this tell us about the direction of motion of the object?
NOTES

36. Calculate the following: Practice!!
A roller coaster moves 85 meters horizontally, then travels 45 meters at an angle of 30.0° above the horizontal. What is its displacement from its starting point?
A pilot sets a plane’s controls, thinking the plane will fly at 2.50 X 102 km/hr to the north. If the wind blows at 75 km/hr toward the southeast, what is the plane’s resultant velocity?

37. Addition of Vectors by Means of Components

38. Calculate the following:
How fast must a truck travel to stay beneath an airplane moving 105 km/hr at an angle of 25° to the ground?
What is the magnitude of the vertical component of the velocity of the airplane in the problem above?

39. Our First Application of Vectors and Components
Name: __________________________ Period: _____ Date: __________
Our First Application of Vectors and Components
We will analyze the motions of “projectiles” using vectors.
Projectiles are objects that are launched or thrown into the air and are subject to gravity.
Projectiles follow parabolic paths.
NOTES

40. Class assignment / Homework
I will not accept late work
Class assignment / Homework

41. Name: __________________________ Period: _____ Date: __________
SECTION: 4
Relative Velocity
NOTES

42. Relative Motion
The motion of an object can be expressed from different points of view
These points of view are known as “frames of reference”
Depending on the frame of reference used, the description of the motion of the object may change

43. Relative Velocity (motion of objects is independent of each other)
Velocity of A relative to B: using the ground as a reference frame
The Language
vAG : v of A with respect to G (ground) = 40 m/s
vBG : v of B with respect to G = 30 m/s
vAB : v of A with respect to B:

44. Example 1
The white speed boat has a velocity of 30km/h,N, and the yellow boat a velocity of 25km/h, N, both with respect to the ground. What is the relative velocity of the white boat with respect to the yellow boat?

45. Example 2- The Bus Ride Lets do this together
A passenger is seated on a bus that is
traveling with a velocity of 5 m/s, North.
If the passenger remains in her seat,
what is her velocity:
with respect to the ground?
with respect to the bus?

46. Example 2 –continued Lets do this together
The passenger decides to approach the driver with a velocity of 1 m/s, N, with respect to the bus, while the bus is moving at 5m/s, N.
What is the velocity of the passenger with respect to the ground?

47. Resultant Velocity (motion of objects is dependent on each other)
The resultant velocity is the net velocity
of an object with respect to a reference
frame.

48. Example 3- Airplane and Wind - relative to the ground
An airplane has a velocity of 40 m/s, N, in still air. It is facing a headwind of 5m/s with respect to the ground.
What is the resultant velocity of the airplane?

49. Frame of reference
What is this guy’s velocity?
What about now?

50. Name: __________________________ Period: _____ Date: __________
Frame of reference
What about now?
NOTES

51. Frame of reference
Definition – a coordinate system from which objects, positions, and velocities are measured. MUST PICK AN ORIGIN before you find speed and velocity.

52. Relative Velocity: Example 4: Crossing a River
The engine of a boat drives it across a river that is 1800m wide.
The velocity of the boat relative to the water is 4.0m/s directed
perpendicular to the current. The velocity of the water relative
to the shore is 2.0m/s.
What is the velocity of the boat
relative to the shore?
B) How long does it take for the boat
to cross the river?

53. Relative Velocity G: U:
E: (Pythagorean Theorem) S:

54. Relative Velocity G: U:
E: (Kinematics) S:

55. Class assignment / Homework
I will not accept late work
Class assignment / Homework