Physics 207: Lecture 4, Pg 1 Lecture 4 l Today: Ch. 3 (all) & Ch. 4 (start)  Perform vector algebra (addition & subtraction) graphically or numerically Interconvert between Cartesian and Polar coordinates  Begin working with 2D motion Distinguish position-time graphs from particle trajectory plots Trajectories  Obtain velocities  Acceleration: Deduce components parallel and perpendicular to the trajectory path Reading Assignment: For Monday read through Chapter 5.3

Physics 207: Lecture 4, Pg 2 Home Exercise: Welcome to Wisconsin l You are traveling on a two lane highway in a car going a speed of 20 m/s. You are notice that a deer that has jumped in front of a car traveling at 40 m/s and that car avoids hitting the deer but does so by moving into your lane! There is a head on collision and your car travels a full 2m before coming to rest. Assuming that your acceleration in the crash is constant. What is your acceleration in terms of the number of g’s (assuming g is 10 m/s 2 )?

Physics 207: Lecture 4, Pg 3 Welcome to Wisconsin l You are traveling on a two lane highway in a car going a speed of 20 m/s (~44 mph). You are notice that a deer that has jumped in front of a car traveling at 40 m/s and that car avoids hitting the deer but does so by moving into your lane! There is a head on collision and your car travels a full 2m before coming to rest. Assuming that your acceleration in the crash is constant. What is your acceleration in terms of the number of g’s (assuming g is 10 m/s 2 )? l Draw a Picture l Key facts (what is important, what is not important) l Attack the problem

Physics 207: Lecture 4, Pg 4 Welcome to Wisconsin l You are traveling on a two lane highway in a car going a speed of 20 m/s. You are notice that a deer that has jumped in front of a car traveling at 40 m/s and that car avoids hitting the deer but does so by moving into your lane! There is a head on collision and your car travels a full 2 m before coming to rest. Assuming that your acceleration in the crash is constant. What is the magnitude of your acceleration in terms of the number of g’s (assuming g is 10 m/s 2 )? l Key facts: v initial = 20 m/s, after 2 m your v = 0.  x = x initial + v initial  t + ½ a  t 2 x - x initial = -2 m = v initial  t + ½ a  t 2  v = v initial + a  t  -v initial /a =  t  -2 m = v initial  -v initial /a  + ½ a (-v initial /a ) 2  -2 m= -½v initial 2 / a  a = (20 m/s) 2 /2 m= 100m/s 2

Physics 207: Lecture 4, Pg 5 Coordinate Systems and vectors l In 1 dimension, only 1 kind of system,  Linear Coordinates (x) +/- l In 2 dimensions there are two commonly used systems,  Cartesian Coordinates(x,y)  Circular Coordinates(r,  ) l In 3 dimensions there are three commonly used systems,  Cartesian Coordinates(x,y,z)  Cylindrical Coordinates (r, ,z)  Spherical Coordinates(r,  )

Physics 207: Lecture 4, Pg 6 Vectors l In 1 dimension, we can specify direction with a + or - sign. l In 2 or 3 dimensions, we need more than a sign to specify the direction of something: r l To illustrate this, consider the position vector r in 2 dimensions. Example Example: Where is Boston ? New York  Choose origin at New York  Choose coordinate system Boston is 212 miles northeast of New York [ in (r,  ) ] OR Boston is 150 miles north and 150 miles east of New York [ in (x,y) ] Boston New York r

Physics 207: Lecture 4, Pg 7 Vectors... l There are two common ways of indicating that something is a vector quantity:  Boldface notation: A  “Arrow” notation: A or A A

Physics 207: Lecture 4, Pg 8 Vectors l Vectors have both magnitude and a direction  Examples: displacement, velocity, acceleration  The magnitude of A ≡ |A|  Usually vectors include units (m, m/s, m/s 2 ) l For vector algebra ( +/ -)a vector has no particular position (Note: the position vector reflects displacement from the origin) l Two vectors are equal if their directions, magnitudes & units match. A B C A = C A = B, B = C

Physics 207: Lecture 4, Pg 9 Scalars l A scalar is an ordinary number.  A magnitude ( can be negative ) without a direction  Often it will have units (i.e. kg) but can be just a number  Usually indicated by a regular letter, no bold face and no arrow on top. Note: Lack of a specific designation can lead to confusion l The product of a vector and a scalar is another vector in the same “direction” but with modified magnitude. A B A = B

Physics 207: Lecture 4, Pg 10 Exercise Vectors and Scalars A. my velocity (3 m/s) B. my acceleration downhill (30 m/s2) C. my destination (the lab - 100,000 m east) D. my mass (150 kg) Which of the following is cannot be a vector ? While I conduct my daily run, several quantities describe my condition

Physics 207: Lecture 4, Pg 11 Vectors and 2D vector addition l The sum of two vectors is another vector. A = B + C B C A B C D = B + 2 C ???

Physics 207: Lecture 4, Pg 12 2D Vector subtraction l Vector subtraction can be defined in terms of addition. B - C B C B -C-C = B + (-1)C A Different direction and magnitude ! A = B + C

Physics 207: Lecture 4, Pg 13 Reference vectors: Unit Vectors l A Unit Vector is a vector having length 1 and no units l It is used to specify a direction. uU l Unit vector u points in the direction of U u  Often denoted with a “hat”: u = û U = |U| û û x y z i j k l Useful examples are the cartesian unit vectors [ i, j, k ] or  Point in the direction of the x, y and z axes. R = r x i + r y j + r z k

Physics 207: Lecture 4, Pg 14 Vector addition using components: l Consider, in 2D, C = A + B. Ciji ji (a) C = (A x i + A y j ) + (B x i + B y j ) = (A x + B x )i + (A y + B y ) Cij (b) C = (C x i + C y j ) l Comparing components of (a) and (b):  C x = A x + B x  C y = A y + B y  |C| =[ (C x ) 2 + (C y ) 2 ] 1/2 C BxBx A ByBy B AxAx AyAy

Physics 207: Lecture 4, Pg 15 Example Vector Addition A. {3,-4,2} B. {4,-2,5} C. {5,-2,4} D. None of the above l Vector A = {0,2,1} l Vector B = {3,0,2} l Vector C = {1,-4,2} What is the resultant vector, D, from adding A+B+C?

Physics 207: Lecture 4, Pg 16 Example Vector Addition A. { A. {3,-4,2} B. {4,-2,5} C. {5,-2,4} D. None of the above l Vector A = {0,2,1} l Vector B = {3,0,2} l Vector C = {1,-4,2} What is the resultant vector, D, from adding A+B+C?

Physics 207: Lecture 4, Pg 17 Converting Coordinate Systems In polar coordinates the vector R = (r,  ) l In Cartesian the vector R = (r x,r y ) = (x,y) We can convert between the two as follows: In 3D cylindrical coordinates (r,  z), r is the same as the magnitude of the vector in the x-y plane [sqrt(x 2 +y 2 )]  tan -1 ( y / x ) y x (x,y)  r ryry rxrx

Physics 207: Lecture 4, Pg 18 Resolving vectors into components A mass on a frictionless inclined plane A block of mass m slides down a frictionless ramp that makes angle  with respect to horizontal. What is its acceleration a ?  m a

Physics 207: Lecture 4, Pg 19 Resolving vectors, little g & the inclined plane g (bold face, vector) can be resolved into its x,y or x ’,y ’ components  g = - g j  g = - g cos  j ’ + g sin  i ’ The bigger the tilt the faster the acceleration….. along the incline x’x’ y’y’ x y g  

Physics 207: Lecture 4, Pg 20 Dynamics II: Motion along a line but with a twist (2D dimensional motion, magnitude and directions) l Particle motions involve a path or trajectory l Recall instantaneous velocity and acceleration l These are vector expressions reflecting x, y & z motion r = r (t)v = d r / dta = d 2 r / dt 2

Physics 207: Lecture 4, Pg 21 Instantaneous Velocity l But how we think about requires knowledge of the path. l The direction of the instantaneous velocity is along a line that is tangent to the path of the particle’s direction of motion. v The magnitude of the instantaneous velocity vector is the speed, s. (Knight uses v) s = (v x 2 + v y 2 + v z ) 1/2

Physics 207: Lecture 4, Pg 22 Average Acceleration l The average acceleration of particle motion reflects changes in the instantaneous velocity vector (divided by the time interval during which that change occurs). l The average acceleration is a vector quantity directed along ∆v ( a vector! )

Physics 207: Lecture 4, Pg 23 Instantaneous Acceleration l The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero l The instantaneous acceleration is a vector with components parallel (tangential) and/or perpendicular (radial) to the tangent of the path l Changes in a particle’s path may produce an acceleration  The magnitude of the velocity vector may change  The direction of the velocity vector may change (Even if the magnitude remains constant)  Both may change simultaneously (depends: path vs time)

Physics 207: Lecture 4, Pg 24 Motion along a path ( displacement, velocity, acceleration ) l 2 or 3D Kinematics : vector equations: rrvrar r = r(  t) v = dr / dt a = d 2 r / dt 2 Velocity : r v = dr / dt r v av =  r /  t Acceleration : v a = dv / dt v a av =  v /  t vvvv y x path

Physics 207: Lecture 4, Pg 25 Generalized motion with non-zero acceleration: need both path & time Two possible options: Change in the magnitude of v Change in the direction of v a = 0 a a a a = + Animation

Physics 207: Lecture 4, Pg 26 Kinematics l The position, velocity, and acceleration of a particle in 3-dimensions can be expressed as: r = x i + y j + z k v = v x i + v y j + v z k (i, j, k unit vectors ) a = a x i + a y j + a z k l All this complexity is hidden away in r = r(  t)v = dr / dta = d 2 r / dt 2

Physics 207: Lecture 4, Pg 27 Special Case Throwing an object with x along the horizontal and y along the vertical. x and y motion both coexist and t is common to both Let g act in the – y direction, v 0x = v 0 and v 0y = 0 y t 0 4 x t = 0 4 y t x 0 4 x vs t y vs t x vs y

Physics 207: Lecture 4, Pg 28 Another trajectory x vs y t = 0 t =10 Can you identify the dynamics in this picture? How many distinct regimes are there? Are v x or v y = 0 ? Is v x >,< or = v y ? y x

Physics 207: Lecture 4, Pg 29 Exercise 1 & 2 Trajectories with acceleration l A rocket is drifting sideways (from left to right) in deep space, with its engine off, from A to B. It is not near any stars or planets or other outside forces. l Its “constant thrust” engine (i.e., acceleration is constant) is fired at point B and left on for 2 seconds in which time the rocket travels from point B to some point C  Sketch the shape of the path from B to C. l At point C the engine is turned off.  Sketch the shape of the path after point C (Note: a = 0)

Physics 207: Lecture 4, Pg 30 Exercise 1 Trajectories with acceleration A. A B. B C. C D. D E. None of these B C B C B C B C A C B D From B to C ?

Physics 207: Lecture 4, Pg 31 Exercise 2 Trajectories with acceleration A. A B. B C. C D. D E. None of these B C B C B C B C A C B D From B to C ?

Physics 207: Lecture 4, Pg 32 Exercise 3 Trajectories with acceleration A. A B. B C. C D. D E. None of these C C C C A C B D After C ?

Physics 207: Lecture 4, Pg 33 Lecture 4 Assignment: Read all of Chapter 4, Ch