Physics 201: Lecture 4, Pg 1 Lecture 4 l Today: Chapter 3  Introduce scalars and vectors  Perform basic vector algebra (addition and subtraction)  Interconvert between Cartesian & Polar coordinates

Physics 201: Lecture 4, Pg 2 An “interesting” 1D motion problem: A race Two cars start out at a speed of 9.8 m/s. One travels along the horizontal at constant velocity while the second follows a 9.8 m long incline angled  below the horizontal down and then an identical incline up to the finish point. The acceleration of the car on the incline is g sin  (because of gravity & little g) At what angles , if any, is the race a tie? 9.8 m   Finish Start

Physics 201: Lecture 4, Pg 3 A race l According to our dynamical equations  x(t) = x i + v i t + ½ a t 2  v(t) = v i + a t A trivial solution is  = 0 ° and the race requires 2 seconds. l Horizontal travel  Constant velocity so t = d / v = 2 x 9.8 m (cos  ) / 9.8 m/s  t 1 = 2 cos  seconds l Incline travel  9.8 m = 9.8 m/s ( t 2 /2) + ½ g sin   t 2 /2) 2  1 = ½ t 2 + ½ sin   (½ t 2 ) 2  sin   t 2 ) t 2  8 =0

Physics 201: Lecture 4, Pg 4 Solving analytically is a challenge A sixth order polynominal, (if z=1, =0)

Physics 201: Lecture 4, Pg 5 Solving graphically is easier Solve graphically  =0.63 rad  = 36° If  < 36° then the incline is faster If  < 36 ° then the horizontal track is faster

Physics 201: Lecture 4, Pg 6 A science project l You drop a bus off the Willis Tower (442 m above the side walk). It so happens that Superman flies by at the same instant you release the bus. Superman is flying down at 35 m/s. l How fast is the bus going when it catches up to Superman?

Physics 201: Lecture 4, Pg 7 A “science” project l You drop a bus off the Willis Tower (442 m above the side walk). It so happens that Superman flies by at the same instant you release the car. Superman is flying down at 35 m/s. l How fast is the bus going when it catches up to Superman? l Draw a picture  y t yiyi 0

Physics 201: Lecture 4, Pg 8 A “science” project l Draw a picture  l Curves intersect at two points y t yiyi 0

Physics 201: Lecture 4, Pg 9 Home Exercise: Welcome to Wisconsin l You are traveling on a two lane highway in a car going a speed of 20 m/s (45 mph). You are notice that a deer that has jumped in front of a car in the opposite lane traveling at 40 m/s (90 mph) 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 201: Lecture 4, Pg 10 Welcome to Wisconsin l You are traveling on a two lane highway in a car going a speed of 20 m/s (~45 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 201: Lecture 4, Pg 11 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 201: Lecture 4, Pg 12 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 201: Lecture 4, Pg 13 Scalars and Vectors l A scalar is an ordinary number.  Has magnitude ( + or - ), but no direction  May have units (e.g. kg) but can be just a number  Represented by an ordinary character Examples: mass (m, M) kilograms distance (d,s) meters spring constant (k) Newtons/meter

Physics 201: Lecture 4, Pg 14 Vectors act like… l Vectors have both magnitude and a direction  Vectors: position, displacement, velocity, acceleration  Magnitude of a vector A ≡ |A| l For vector addition or subtraction we can shift vector position at will (NO ROTATION) l Two vectors are equal if their directions, magnitudes & units match. A B C A = C A ≠B, B ≠ C

Physics 201: Lecture 4, Pg 15 Vectors look like... l There are two common ways of indicating that something is a vector quantity:  Boldface notation: A  “Arrow” notation: A or A A

Physics 201: Lecture 4, Pg 16 Scalars and Vectors l A scalar can’t be added to a vector, even if they have the same units. 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 201: Lecture 4, Pg 17 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 201: Lecture 4, Pg 18 Vectors and 2D vector addition l The sum of two vectors is another vector. A = B + C B C A B C

Physics 201: Lecture 4, Pg 19 2D Vector subtraction l Vector subtraction can be defined in terms of addition. B - C B -C B -C-C B - C = B + (-1)C B+C Different direction and magnitude !

Physics 201: Lecture 4, Pg 20 Unit Vectors l A Unit Vector points : a length 1 and no units l Gives a direction. uU l Unit vector u points in the direction of 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 or R = x i + y j + z k

Physics 201: Lecture 4, Pg 21 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 201: Lecture 4, Pg 22 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 201: Lecture 4, Pg 23 Converting Coordinate Systems (Decomposing vectors) 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  tan -1 ( y / x ) y x (x,y)  r ryry rxrx

Physics 201: Lecture 4, Pg 24 Motion in 2 or 3 dimensions l Position l Displacement l Velocity (avg.) l Acceleration (avg.)