1. Physics 218
Lecture 2:
Units and Vectors
Kinematics
Alexei Safonov

2. Checklist Yesterday: Wednesday: Sunday:
Homework for Chapter 1 submitted via Mastering
Pre-lectures and check-points completed before 8:00AM today
Wednesday:
Pre-lectures and checkpoints for 1-Dim motion
Sunday:
Homework for Chapter 2 due

3. Today Talk more about vectors Second half of the lecture:
Operations with vectors: scalar and vector products
Will also revisit some of the checkpoints from the pre-lectures
Second half of the lecture:
Kinematics in 1 dimensional case

4. Clickers Setup Turn on your clicker (press the power button)
Set the frequency:
Press and hold the power button
Two letters will be flashing
If it’s not “BD”, press “B” and then “D”
If everything works, you should see “Welcome” and “Ready”

5. Clicker Question 1 We use the “BD” frequency in this class
Do you have your i>clicker with you today?
Yes No
Maybe
I like pudding
We use the “BD” frequency in this class

6. Math Prelecture Math Preview: View this as a test of your math skills:
Most people got through it fine
Most concerns were about the “sine theorem”
View this as a test of your math skills:
Take action to catch up in the areas which this review found to be problematic for you

7. Pre-Lecture Question
You look it up and find that there are 2.54 centimetres in one inch
The motorcycle engine on a Kawasaki Ninja 1000 has a displacement of 1043 cubic-centimeters (cm3). In order to calculate its engine displacement in cubic-inches (in3) what unit conversion factor would you use to multiply the given displacement?
A. 1 in3 / 2.54 cm3
B cm3 / 1 in3
C. 1 in3 / 16.4 cm3
D cm3 / 1 in3

8. Pre-Lecture Question
You look it up and find that there are 2.54 centimeters in one inch
The motorcycle engine on a Kawasaki Ninja 1000 has a displacement of 1043 cubic-centimeters (cm3). In order to calculate its engine displacement in cubic-inches (in3) what unit conversion factor would you use to multiply the given displacement?
A. 1 in3 / 2.54 cm3
B cm3 / 1 in3
C. 1 in3 / 16.4 cm3
D cm3 / 1 in3

9. Specifying a Vector
Two equivalent ways:
Components Vx and Vy
Magnitude V and angle q
Switch back and forth
Magnitude of V
|V| = (vx2 + vy2)½
Pythagorean Theorem
tanq = vy /vx
Either method is fine, pick one that is easiest for you, but be able to use both

10. Unit Vectors ^ Another notation for vectors: ^ ^ kx z y ^ k
Another notation for vectors:
Unit Vectors denoted i, j, k ^ i ^ j

11. Unit Vectors
Similar notations, but with x, y, z x z y ^ j i k

12. Vector in Unit Vector Notation

13. General Addition Example
Add two vectors using the i-hats, j-hats and k-hats

14. Simple Multiplication
Multiplication of a vector by a scalar
Let’s say I travel 1 km east. What if I had gone 4 times as far in the same direction?
→Just stretch it out, multiply the magnitudes
Negatives:
Multiplying by a negative number turns the vector around

15. Subtraction Subtraction is easy:
It’s the same as addition but turning around one of the vectors. I.e., making a negative vector is the equivalent of making the head the tail and vice versa. Then add:

16. Vector Question
Vector A has a magnitude of 3.00 and is directed parallel to the negative y-axis and vector B has a magnitude of 3.00 and is directed parallel to the positive y-axis. Determine the magnitude and direction angle (as measured counterclockwise from the positive x-axis) of vector C, if C=A−B.
A. C=0.00 (its direction is undefined) B. C = 3.00; θ = 270o C. C = 3.00; θ = 90o D. C = 6.00; θ = 270o F. C = 6.00; θ = 90o

17. Vector Question
Vector A has a magnitude of 3.00 and is directed parallel to the negative y-axis and vector B has a magnitude of 3.00 and is directed parallel to the positive y-axis. Determine the magnitude and direction angle (as measured counterclockwise from the positive x-axis) of vector C, if C=A−B.
A. C=0.00 (its direction is undefined) B. C = 3.00; θ = 270o C. C = 3.00; θ = 90o D. C = 6.00; θ = 270o F. C = 6.00; θ = 90o

18. How do we Multiply Vectors?
First way: Scalar Product or Dot Product
Why Scalar Product?
Because the result is a scalar (just a number)
Why a Dot Product?
Because we use the notation A.B
A.B = |A||B|CosQ

19. First Question:
A.B = |A||B|CosQ x z y ^ j i k

20. First Question:
A.B = |A||B|CosQ 1 -1
unit vector k x z y ^ j i k

21. Harder Example

22. Vector Cross Product
This is the last way of multiplying vectors we will see
Direction from the “right-hand rule”
Swing from A into B!

23. Vector Cross Product Cont…
Multiply out, but use the Sinq to give the magnitude, and RHR to give the direction x z y ^ j i k j k i + i k j _

24. Cross Product Example

25. Vector Product Calculate the vector product C=A x B:
Bold font means vector (same as having an arrow on the top)
Vector A points in positive y direction and has magnitude of 3
Vector B points in negative x direction and has magnitude of 3
Which is the correct way to calculate C?
A. C = 3j x (-3i) = - 9k
B. C = 3j x (-3i) = +9k
C. C = 3 x (-3) x sin (90o) = - 9
D. C = 3 x (-3) x sin (270o) = + 9

26. Scalar Product Calculate the scalar product A⋅B: 11.6 12.0 14.9 15.4
19.5

27. Vector Product Calculate the scalar product A⋅B: 11.6 into the page
11.6 out of the page
12.0 into the page
12.0 out of the page
14.9 into the page
14.9 out of the page
15.4 into the page
15.4 out of the page

28. Kinematics in 1 dimension

29. Kinematics: Describing Motion
Interested in two key ideas:
How objects move as a function of time
Kinematics
Chapters 2 and 3
Why objects move the way they do
Dynamics
Do this in Chapter 4 and later

30. Chapter 2: Motion in 1-Dimension
Velocity & Acceleration
Equations of Motion
Definitions
Some calculus (derivatives)
Wednesday:
More calculus (integrals)
Problems

31. Notes before we begin
This chapter is a good example of a set of material that is best learned by doing examples
We’ll do some examples today
Lots more next time…

32. Lecture Thoughts from FIP
I'm used to using math-based approaches to find velocity, acceleration, and position, so understanding how to use physics, and why we should use physics instead of simply deriving and integrating is fuzzy to me
All physics problems are math problems with boundary conditions. You need to understand physics to correctly set boundary conditions, so you have a well defined math problem. Then it’s all math.
Questions of reading, understanding and interpreting graphs and their relationship with formulas:
Lots and lots of questions, so we will heavily focus on that today

33. We want Equations that describe:
Equations of Motion
We want Equations that describe:
Where am I as a function of time?
How fast am I moving as a function of time?
What direction am I moving as a function of time?
Is my velocity changing? Etc.

34. Motion in One Dimension
Where is the car?
X=0 feet at t0=0 sec
X=22 feet at t1=1 sec
X=44 feet at t2=2 sec
We say this car has “velocity” or “Speed”
Plot position vs. time. How do we get the velocity from the graph?

35. Motion in One Dimension Cont…
Velocity: “Change in position during a certain amount of time”
Calculate from the Slope: The “Change in position as a function of time”
Change in Vertical
Change in Horizontal
Change: D
Velocity  DX/Dt

36. Constant Velocity Equation of Motion for this example: X = bt
Slope is constant
Velocity is constant
Easy to calculate
Same everywhere