1. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Chapter 4: Motion in 2 and 3 dimensions To introduce the concepts and notation for vectors: quantities that, in a single package, convey magnitude and direction motion To generalize our 1d kinematics to higher numbers of dimensions, using vector notation To generalize Newtonian dynamics using vectors To appreciate the glory of the Free Body Diagram and how it enables one to utilize N2 To discuss projectile motion, and motion on a curve Chapter 4 Goals:

2. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Polar coordinates in 2 dimensions The position of a point (x,y) are the cartesian coordinates two numbers needed to specify full information convention for  is to start along x and swing counterclockwise r   hypotenuse x   adjacent y   opposite

3. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Polar coordinates in 2 dimensions r   hypotenuse x   adjacent y   opposite h o a

4. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The displacement vector: we move into a higher dimension at last! initial position is r i final position is r f displacement is  r:= r f – r i x riri rfrf rr Notice : the positions are origin-dependent, but the displacement is not!! y If it behaves like the displacement vector, then it is a vector!! It is the ‘paradigmatic’ vector. vector subtraction is a bit tricky

5. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Obvious language for vectors in two dimensions: magnitude and direction x y  \A\ A magnitude of A: written |A| (some books say A) it is just the length of A |A| is always positive or 0 technically it is not a scalar but don’t worry about that |A| carries the units of A direction of A: written  A usually counterclockwise from x axis units of  A are degrees or radians

6. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Pros of magnitude-direction language we can write A = {|A|,  A } lends itself to obvious pictures can easily be converted to compass (map) language: East   x, and North   y addition and subtraction of vectors is done pictorially accuracy of pictorial addition and subtraction is limited to human ability laws of sines and cosines needed to calculate: messy only convenient in 2d!! 3d requires the dreaded spherical trigonometry because there are 2 angles!! Cons of magnitude-direction language

7. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Triangle method for adding vectors A (and B) are two displacement vectors with B following A magnitude : |A| or A direction: an angle , just like in previous figure A is followed by B, to give resultant R = A + B ‘tip-to-tail’  triangle method of adding vectors One can just keep going, tip to tail style

8. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Rules for adding vectors (and to multiply by scalar) A + B = B + A: vector addition is commutative (A + B) + C = A + (B + C): vector addition is associative the negative of a vector is – A: same length as A but opposite direction [additive inverse] There is a zero vector (no length) vectors carry units and when added units mus be the same ‘tip-to-tail’  triangle method of addition cA: also a vector (opposite direction) if c<0 but ‘rescaled’ in length: |cA| = |c| |A|

9. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. How to subtract one vector from another A – B := A + (– B): to subtract, just add the additive inverse put both A and B tail-to-tail: then C = A – B has its tail at the tip of B (the one subtracted) and has its tip at the tip of A (the one added) it seems to ‘start’ at B and end at A This procedure is extremely important when finding changes in vectors!!! Example: the displacement vector is precisely the change in the position vector!! {show Active Figure AF_0306

10. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Addition Example : Taking a Hike A hiker begins a trip by first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. On the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower. Using ruler and protractor, lay out the two displacements A and B. Then, with the same tools, measure and then

11. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. A different and more flexible language: components x y  |A| A AxAx AyAy in this sketch,  A is larger than 90 but you can deal… |A| is the hypotenuse; |A| ≥ 0 A x is the adjacent and here < 0 A y is the opposite and here > 0 “drop a perpendicular” from tip to either Cartesian axis you have made a right triangle draw vector with tail at origin: standard position we write A =

12. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Converting from magnitude-angle to components This assumes that the angle is defined as ccw from x axis Pros of component language addition and subtraction of two (or more) vectors is simplicity itself: just add (subtract) the components like scalars and the resultant vector‘s (difference vector’s) components are known! {show Active Figure AF_0303}

13. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Revisiting the vector addition example Section 3.4

14. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Finishing the Example Section 3.4 Now we would work back from component language to magnitude-angle language: ‘net displacement is 41.3 km, at a bearing of 65.9° East of North’ R = {41.3 km, 24.1°}

15. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. A third language that uses components: unit vectors x y  |A| A i is a vector of unit length, with no units, that points along x j and k are similar, along y (and z) create the vectors i A x & j A y j A y i A x A truly explicit way to write A remember: |i| = |j| = |k| = 1 (no units, and but one unit long!!)

16. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. What other operations can we do with vectors? cannot divide by a vector; vectors are only ‘upstairs’ dot (scalar (inner)) product of two cross (vector (outer (wedge))) product of two we can take their derivative with respect to a scalar we can integrate them but usually we integrate some kind of scalar product… The scalar product of two vectors A B  put them tail-to-tail, with  the angle between (0° ≤  ≤ 180°) the result is indeed a scalar A∙B = B ∙A (commutative)

17. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. More about the scalar product a measure of the parallelness of the vectors, as well as the magnitudes A∙B = (|A|)(|B| cos  ) = length of A times the length of B’s projection along the line of A A∙B = (|B|)(|A| cos  ) = length of B times the length of A’s projection along the line of B A∙A = |A| 2 i∙i = j∙j = k∙k =1 i∙j = j∙k = 0 etc. a vector’s component in a certain direction is the scalar product of that vector with a unit vector in that direction: C n = C∙n [to make a unit vector, just divide a vector by its magnitude: a = A/|A| ]

18. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Kinematics in ‘higher dimensions’ I: position and displacement as vector position is now a vector: the three components of the position: r:= if the arena is 2d, drop the z-stuff displacement is change in position:  r = r B – r A or  r = r f – r i or, best of all,  r = r(t+  t) – r(t) it is a vector with tail at r(t) and tip at r(t+  t) displacement vector not usually drawn in standard position but may be, especially if you are adding a second displacement to the first and you don’t really care about ‘initial’ position, or you take that to be zero the displacement vector is origin-independent!

19. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Kinematics in ‘higher dimensions’ II: the (instantaneous) velocity vector here we used the fact that the unit vectors do not vary with time as the body moves around the three components of the velocity v:= where v x (t)=dx(t)/dt and similarly for y and z velocity is now a vector:

20. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Kinematics in ‘higher dimensions’ III x y r(t)r(t) r(t+t)r(t+t) r(t)r(t) path of object how do we understand this? start with average velocity: of course, =  r/  t is a vector: it’s a vector (  r) times a number (1/  t) magnitude: | | = |  r |/|  t| direction: same as direction of  r

21. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Kinematics in ‘higher dimensions’ IV x y r(t)r(t) r(t+t)r(t+t) r(t)r(t) path of object we now let  t get really small (  t  0) and call it dt as that shrinks, so does  r  : call it dr r(t+dt) dr(t)dr(t) direction: tangent to the path in space!!! magnitude: |v| = |dr |/dt| = speed