Physics 207: Lecture 4, Pg 1 Lecture 4 l Goals for Chapter 3 & 4  Perform vector algebra (addition & subtraction) graphically or by xyz components 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  Solve classic problems with acceleration(s) in 2D (including linear, projectile and circular motion) motion in stationary and moving frames  Discern different reference frames and understand how they relate to motion in stationary and moving frames Assignment: Read thru Chapter 4 MP Problem Set 2 MP Problem Set 2 due this coming Wednesday

Physics 207: Lecture 4, Pg 2 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 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 6 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 7 Reference vectors: Unit Vectors Unit Vector 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 i, j, k l Useful examples are the cartesian unit vectors [ i, j, k ]  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 8 Vector addition using components: CAB 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 9 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 10 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 11 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 12 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 13 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 14 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 15 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 16 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 17 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 18 Lecture 4 Assignment: Read through Chapter 4 MP Problem Set 2 due Wednesday