Physics 218, Lecture V1 Physics 218 Lecture 5 Dr. David Toback

Physics 218, Lecture V2

3 Chapter 3 Kinematics in Two Dimensions Vectors

Physics 218, Lecture V4 Overview Motion in multiple dimensions Vectors: (Tools to solve problems) –Why we care about them –Addition, Subtraction and Multiplication Graphical and Component –Unit Vectors Projectile Motion (Next lecture) –Problem Solving

Physics 218, Lecture V5 Why do we care about Vectors? Last week we worked in one dimension However, as you may have noticed, the world is not one-dimensional. Three dimensions: X, Y and Z. Example: 1.Up from us 2.Straight in front of us 3.To the side from us –All at 90 degrees from each other. Three dimensional axis. –Need a way of saying how much in each direction For this we use VECTORS

Physics 218, Lecture V6 Another reason to care about vectors It turns out that nature has decided that the directions don’t really care about each other. Example: You have a position in X, Y and Z. If you have a non-zero velocity in only the Y direction, then only your Y position changes. The X and Z directions could care less. (I.e, they don’t change). Represent these ideas with Vectors

Physics 218, Lecture V7 Vector notation: –In the book, variables which are vectors are in bold –On the overheads, I’ll use an arrow over it Vectors are REALLY important Kinda like calculus: These are the tools! First the Math: Vector Notation Some motion represented by vectors. What do these vectors represent physically?

Physics 218, Lecture V8 Vector Addition Why might I need to add vectors? –If I travel East for 10 km and then North for 4 km, it would be good to know where I am. What is my new position? Can think of this graphically or via components –Graphically: Lay down first vector (the first part of my trip) Lay down second vector (the second part of the trip) with its tail at the head of the first vector The “Sum” is the vector from the tail of the first to the head of the second First Second Sum Adding vectors is a skill Use this in far more than just physics

Physics 218, Lecture V9 Examples without an axis

Physics 218, Lecture V10 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

Physics 218, Lecture V11 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:

Physics 218, Lecture V12 The tricky part We saw that if you travel East for 10 km and then North for 4 km, you end up with the same displacement as if you traveled in a straight line NorthEast. Could think of this the other way: If I had gone NorthEast, it’s the equivalent of having gone both North and East. My single vector in some funny direction, can be thought of as two vectors in nice simple directions (like X and Y). This makes things much easier.

Physics 218, Lecture V13 Components This is the tricky part that separates the good students from the poor students Break a vector into x and y components (I.e, a right triangle) THEN add them This is the sine and cosine game Can use the Pythagorean Theorem: A 2 + B 2 = C 2 Again, this is a skill. Get good at this!!!

Physics 218, Lecture V14 Adding Vectors by Components How do you do it? First RESOLVE the vector by its components! Turn one vector into two V = V x + V y V x = Vcos  V y = Vsin  Careful when using the sin and cos

Physics 218, Lecture V15 Specifying a Vector Two equivalent ways: –Components V x and V y –Magnitude V and angle  Switch back and forth –Magnitude of V |V| = (v x 2 + v y 2 ) ½ (Pythagorean Theorem) –Tan  = v y /v x

Physics 218, Lecture V16 Example What is the magnitude and angle of the displacement in this example?

Physics 218, Lecture V17 Adding vectors in funny directions Let’s say I walk in some random direction, then in another different direction. How do I find my total displacement? We can draw it It would be good to have a better way…

Physics 218, Lecture V18 Addition using Components This is the first half of how pros do it: To add two vectors, break both up into their X and Y components, then add separately Magnitudes

Physics 218, Lecture V19 Drawing the components

Physics 218, Lecture V20 Unit Vectors This is how the pros do it!

Physics 218, Lecture V21 Simple Example What is the displacement using Unit Vectors in this example?

Physics 218, Lecture V22 Example: Adding Unit Vectors

Physics 218, Lecture V23 Mail Carrier and Unit Vectors A rural mail carrier leaves leaves the post office and drives D 1 miles in a Northerly direction to the next town. She then drives in a direction  South of East for D 2 miles to another town. Using unit vector notation, what is her displacement from the post office?

Physics 218, Lecture V24 Vector Kinematics Continued

Physics 218, Lecture V25 Constant Acceleration

Physics 218, Lecture V26 Projectile Motion This is what all the setup has been for! Motion in two dimensions –For now we’ll ignore air friction

Physics 218, Lecture V27 Projectile Motion The physics of the universe: The horizontal and vertical parts of the motion behave independently This is why we use vectors in the first place

Physics 218, Lecture V28 Ball Dropping Analyze Vertical and Horizontal separately!!! A y = g (downwards) A x = 0 –Constant for Both cases!!! V x = 0V x >0

Physics 218, Lecture V29 A weird consequence An object projected horizontally will reach the ground at the same time as an object dropped vertically. Proof:

Physics 218, Lecture V30 Rest of this week Reading: Finish Chapter 3 if you haven’t already Homework: –Finish HW2 and be working on HW3 Web quiz: If you don’t have ten 100’s yet, I recommend you do so before the exam (coming up!) Labs and Recitations: Both meet this week. Next time: More on kinematics in two dimensions and vectors

Physics 218, Lecture V31 A Mail Carrier A rural mail carrier leaves leaves the post office and drives D 1 miles in a Northerly direction to the next town. She then drives in a direction  degrees South of East for a distance D 2 to another town. What is the magnitude and angle of her displacement from the post office?

Physics 218, Lecture V32 Vector stuff 1.Pythagorean theorem: We’ll use this a lot –For a right triangle (90 degrees) –Length C is the hypotenuse –A 2 + B 2 = C 2 2.Vector equations

Physics 218, Lecture V33 Using all this stuff