VECTOR ARITHMETIC

https://www.youtube.com/watch?v=pyBNImQkRuk

FRAME OF REFERENCE

STEWARDESS She just sees the juice fall straight down into the cup 9.8 m/s2. Nothing strange.

But Don’t Forget. . . . . . . The cup is also shooting forward 400 miles per hour. I want to see YOU throw something into a moving cup! Good luck!

What actually happens The juice moves FORWARD 400 miles per hour while accelerating DOWNWARD 9.8 m/s2. It results in a beautiful arc right into the cup.

Who is right? The Stewardess who says it falls straight down? Or the people on the ground that say it was moving forward 400 miles per hour? We are BOTH right!!! What we see just depends on our FRAME OF REFERENCE.

FRAME OF REFERENCE A frame of reference is what you are comparing your results with. For example, if we use the inside of the airplane as our frame of reference, then the orange juice fell straight down. But if we use the surface of the Earth as our frame of reference then the orange juice is moving forward 400 miles per hour.

NEWTONIAN FRAME OF REFERENCE Sir Isaac Newton said that any frame of reference is “valid” as long as it (the reference frame) is not accelerating. If you are inside a Newtonian Frame of Reference then all the laws of physics will appear to be obeyed and everything will act “normal”.

https://www.youtube.com/watch?v=3ug23VTMies

The PING PONG Question If you could play a normal game of ping pong inside of your frame of reference, then it is a “valid” Newtonian Frame of Reference.

Am I Valid? I am in an airplane moving 400 miles per hour? YES. As long as the plane doesn’t speed up, slow down, or turn, then it is a valid frame of reference. If you are inside the plane everything will obey the laws of physics. You could play a game of ping pong and nothing “strange” would happen.

Am I valid? I am in the back of a semi-truck driving around a circular race track? NO! Turning counts as acceleration. If you tried to play ping pong in the back of the truck “weird” stuff would happen. The ball would keep veering towards the outside wall of the truck even if you didn’t hit it that direction.

RELATIVITY If a measurement changes depending on your point of view, then we say it is “RELATIVE”. For example, how fast the car next to you on the freeway appears to be moving depends on how fast YOU yourself are moving. If you match speeds they “appear” to be parked next to you.

RELATIVITY When NASA repairs satellites they first have to find the satellite and then match its speed. The astronaut doing the repairs is actually hurtling through space hundreds of miles an hour. But the satellite “appears” to hold still because the astronaut is moving with it.

RELATIVITY In old movies it was the SCENERY that moved and not the cars. Easier to film that way. Your brain can’t tell the difference.

RELATIVITY So far this year we have learned two important values that are “relative”. POSITION and VELOCITY: Where is the orange juice now? And where is it going next? The Stewardess might say the position is not changing because it stays inside the glass. Someone on the ground might say the position is changing quickly as the cup shoots forward across the sky. Both are right.

ABSOLUTE Not everything changes depending on your frame of reference. Some things stay the same no matter what. We call them “absolutes”.

ABSOLUTE What was the ONE thing about the orange juice that everybody agrees on? It fell into the cup 9.8 m/s2. The reason we agree about this is because ACCELERATION is ABSOLUTE. No matter who measures it or what their own frame of reference is doing, everyone will observe the same behavior and obtain the same numeric answers.

POINT OF VIEW Our own personal bias must be taken into account whenever we measure something. What we observe is called our “Point of View”. For example, when tall people walk down the hall everyone appears to be little. When short people walk down the hall everyone appears to be big. It just depends on your POINT OF VIEW!!

POINT OF VIEW Imagine that you are walking down the hall throwing a tennis ball up and then catching it. Let’s look at what is happening in more detail.

MY POINT OF VIEW You will see the ball leave your hand going straight up and then fall back down into your hand coming straight down. As long as you walk at a constant speed nothing “strange” should happen.

OTHERS POINT OF VIEW The tennis ball moves forward with you in a parabolic arc.

OTHERS POINT OF VIEW http://www.youtube.com/watch?v=mYH_nODWkqk

SUBTRACTION Subtraction tells us how DIFFERENT two numbers are from each other. How different is the number 10 from the number 7? RELATIVE VALUE = 10 – 7 = 3 From 7’s point of view, the number 10 is THREE units bigger.

VECTOR SUBTRACTION Vector subtraction tells us how DIFFERENT two vectors are from each other. How different is the velocity of a hungry shark compared to my own velocity? (In other words, am I going to survive this!) RELATIVE VELOCITY = shark vector – my vector From my point of view the shark is moving ____ m/s.

ORDER MATTERS 10 – 7 = 3 7 – 10 = -3 Clearly the ORDER MATTERS What is Seen = Observed - Observer

WHEN DO I USE VECTOR SUBTRACTION If there are TWO objects in the story that are being defined by TWO different vectors. Example: Bird A is flying North 10 m/s and Bird B is flying East 20 m/s.

VECTOR SUBTRACTION 1. STEP 1: Draw a DOT 2. STEP 2: Draw BOTH vectors starting on the dot 3. STEP 3: Connect the observER tip to the observED tip. This shows you how DIFFERENT the vectors are. I like to put eyeballs on the observer to help me do this correctly. The new vector you drew is the answer. It should have both a SIZE and a DIRECTION.

EXAMPLE Imagine that Bob is walking 8 m/s to the right. His little sister Susy can’t keep up. She is walking 3 m/s to the right. What is happening from SUSY’S point of view? Bob’s Vector = 8 Notice the resultant vector points forward and has a length of 5 m/s. Susy will see Bob moving forward 5 m/s. Susy’s Vector = 3 This is what she sees. Resultant = 5 Susy is the observer.

EXAMPLE What is happening from BOB’S point of view? Bob is the observer. Notice the resultant vector points backward and has a length of 5 m/s. Bob will see Susy moving backward 5 m/s. His brain knows she isn’t actually walking backwards. But his eyes see her getting further and further behind him. Bob’s Vector = 8 Susy’s Vector = 3 This is what he sees. Resultant = back 5

EXAMPLE A is the observer. Train A is driving North 57 mph. Train B is driving South 65 mph. A passenger in train A looks out the window. How fast will train B appear to be approaching? (This is why head on collisions are so dangerous.) Train A = 57 This is what he sees. Resultant = south 122 Train B = 65 Notice the resultant vector points to the south going 122 mph. If those trains got in a head on collision the RELATIVE speed would be over a hundred miles an hour! Yikes!

NOTICE In the previous example I drew the arrows going away from each other even though the story said the trains were approaching each other. That is because for vector subtraction you start both arrows from the same point! Even if the objects didn’t actually start together in real life.

where c is the hypotenuse TWO DIMENSIONS The technique works in two dimensions. Just follow the three steps for Vector Subtraction and you will be fine! It also helps to remember the pythagorean theorem! a2 + b2 = c2 where c is the hypotenuse

EXAMPLE Airplane A is flying North 120 mph. Airplane B is flying East 240 mph. A passenger in plane B looks out the window. How fast (and in what direction) will plane A appear to be flying? This is what he sees. Resultant = pythagorean theorem = 268 Plane A = 120 Notice the resultant vector points to the Northwest going 268 mph. That is what passenger B will see from his window. B is the observer. Plane B = 240