Physics 218 Lecture 2: Units and Vectors Kinematics Alexei Safonov

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

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

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”

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

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

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. 2.54 cm3 / 1 in3 C. 1 in3 / 16.4 cm3 D. 16.4 cm3 / 1 in3

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. 2.54 cm3 / 1 in3 C. 1 in3 / 16.4 cm3 D. 16.4 cm3 / 1 in3

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

Unit Vectors ^ Another notation for vectors: ^ ^ k x z y ^ k Another notation for vectors: Unit Vectors denoted i, j, k ^ i ^ j

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

Vector in Unit Vector Notation

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

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

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:

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

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

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

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

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

Harder Example

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!

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 _

Cross Product Example

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

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

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

Kinematics in 1 dimension

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

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

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…

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

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.

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?

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

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