This Week 6/21 Lecture – Chapter 3 6/22 Recitation – Spelunk Problems: 3.3, 3.29, 3.31 6/23 Lab – Kinematics in 1-D 6/24 Lecture – Chapter 4 6/25 Recitation – Projectile Motion Problems: 4.7, 4.18, 4.28, 4.59 6/21/04

Chapter 3 Vectors

Vectors Vectors have a magnitude and a direction In 1 dimension we could specify direction with +/- In 2 and 3 dimensions we need more Columbus to Harrisburg 315 mi and 5o wrt East Columbus to Nashville 315 mi and 235o wrt East 6/21/04

Notational Convention How do we know when we are talking about a vector quantity? Arrow A Boldface A Underline A 6/21/04

Representation of Vectors Graphical Polar Cartesian Magnitude Direction x and y components Use the representation most appropriate to the problem 6/21/04

Adding Vectors (Graphically) Vectors are added “Tail to Tip” 6/21/04

Trigonometry Review Sin q = opp/hyp Cos q = adj/hyp Tan q = opp/adj hypotenuse opposite side adjacent side Sin q = opp/hyp Cos q = adj/hyp Tan q = opp/adj Mnemonic: Soh Cah Toa Note: p radians = 180o 6/21/04

Unit Vectors Carry the “direction” information Unit magnitude Dimensionless velocity vector x component of velocity vector unit vector in the x direction 6/21/04

Converting from Polar to Cartesian q r Given: Magnitude: r Direction: q x component of the vector y component of the vector 6/21/04

Converting from Cartesian to Polar Given rx, ry: q r Pythagorean Theorem Geometry Caution: Quadrant Ambiguity! 6/21/04

Example: Find the magnitude and direction of a vector given by: v = (vx ,vy) = (-5,-3) m/s y x v vy=-3m/s f vx=-5m/s Note that f is in the -x,-y quadrant, so q = 59° +180° =239° 6/21/04

Example: Given the following velocity, calculate the acceleration 6/21/04

Vector Addition (Quantitative) Given two vectors Add the components to get the vector sum 6/21/04

Vector Addition We can verify this equation graphically x y Ax Bx By Ay 6/21/04

Example: Find the vector sum of the following: 6/21/04

Scalar multiplication Can multiply a vector by a scalar: Example: 6/21/04

Properties of Vectors Addition Commutative Property Associative Property Distributive Property Distributive Property 6/21/04

Properties of Vectors Associative Property Commutative Property 6/21/04

Multiplication by a scalar Vector Properties Multiplication by a scalar If c is positive: Just changes the length of the vector If c is negative: Length of vector changes Direction changes by 180° 6/21/04

Dot or Scalar Product f is the angle between the vectors if you put their tails together B f A Recall that cos(f) = cos(-f), so 6/21/04

Dot Product: Physical Meaning Measures “how much” one vector lies along another B f A 6/21/04

Example: Find the angle f between vectors A and B: First, solve for f: 6/21/04

Solution: Plugging these into our equation for f: 6/21/04

“Cross” or Vector Product Another way to multiply vectors… Magnitude only… 6/21/04

Example: Find the cross product of A and B Only in the xy-plane Perpendicular to the xy-plane 6/21/04

Right Hand Rule Point your fingers in the direction of the first vector Curl them in the direction of the second vector Your thumb now points in the direction of the cross product Sanity check your answers! 6/21/04

Multiplication of Unit Vectors Can multiply by components, but it usually takes longer and is easier to make mistakes + 6/21/04

Properties of the Cross Product Cross product normal to the surface of the plane created by A and B Antisymmetric http://physics.syr.edu/courses/java-suite/crosspro.html 6/21/04

Properties of Vector Products Antisymmetry Multiplication by a scalar Distributive property Distributive property Triple Product Triple Product 6/21/04

Vector Multiplication Dot Product result is a scalar projection of one vector onto the other Cross Product result is a vector resultant vector is perpendicular to both vectors 6/21/04

Where to Shoot? a) Aim at target b) Aim ahead of target vc a) Aim at target b) Aim ahead of target c) Aim behind target 6/21/04

Where to Shoot? If Vc=15 m/s, Va=50 m/s, what is θ? sinθ=Vc/Va Vnet θ Vnet = Vc + Va If Vc=15 m/s, Va=50 m/s, what is θ? Vnet vc Va θ sinθ=Vc/Va =15/50=0.3  θ=17.5° What is Vnet? Vnet = Va cosθ = 50 cos(17.5)° = 47.7 m/s If the arena diameter is 60m, how long is arrow flight? Radius is 30m, time of flight is: Ta=R/Vnet=(30 m)/(47.7 m/s)=0.63s 6/21/04

Example (Problem 3.11) A woman walks 250 m in the direction 30 east of north, then 175 m directly east. Find the magnitude of her final displacement, the angle of her final displacement, and the distance she walks. Which is greater, that distance or the magnitude of her displacement? 6/21/04

Chapter 4 Motion in 2-D and 3-D

Chapter 4: 2D and 3D Motion Ideas from 1D kinematics can be carried over into 2D and 3D. Our kinematic variables are now vectors. 6/21/04

Simplest Quantity: Position r(t) Position of particle is specified by r(t) which is a vector depending on time Separate into components r(t)=x(t) î + y(t) ĵ 6/21/04

Displacement In 1-D In 2-D 6/21/04

Velocity Just like 1-D: Average Velocity Instantaneous Velocity y(t) x(t) Direction is tangent to the path 6/21/04

Velocity (4m, 3m) (5m, 2m) Example: Say an object moves from point 1 to point 2 in 1 s. Find the average velocity of the object: Two 1-D Problems 6/21/04

Acceleration Just like 1-D: Average Acceleration vi vf a Instantaneous Acceleration 6/21/04

Treat each dimension separately In x-direction: x, vx = dx/dt, and ax = dvx/dt In y-direction: y, vy = dy/dt, and ay = dvy/dt In z-direction: z, vz = dz/dt, and az = dvz/dt Like three one-dimensional problems 6/21/04

a and v Follow From r(t) v a r Note: r, a, and v don’t have to point in the same direction! 6/21/04

Example: (Problem 4.2) The position vector for an electron is r = (5.0 m)i – (3.0 m)j – (2.0 m)k Find the magnitude of r Sketch the vector on a right handed coord system ^ ^ ^ 6/21/04

Example: (Problem 4.9) A particle moves so that its position (in meters) as a function of time (in seconds) is r = i + 4 t2 j + t k. Write expressions for its velocity as a function of time and its acceleration as a function of time. 6/21/04