Lecture 4

1. Lecture 4 • 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) • 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 due this coming Wednesday

2. Welcome to Wisconsin • 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/s2)?

3. Welcome to Wisconsin • 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/s2)? • Draw a Picture • Key facts (what is important, what is not important) • Attack the problem

4. Welcome to Wisconsin • 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/s2)? • Key facts: vinitial = 20 m/s, after 2 m your v = 0. • x = xinitial + vinitialDt + ½ a Dt2 x - xinitial = -2 m = vinitialDt + ½ a Dt2 • v = vinitial + a Dt  -vinitial /a = Dt • -2 m = vinitial(-vinitial /a ) + ½ a (-vinitial /a )2 • -2 m= -½vinitial 2 / a  a = (20 m/s) 2/2 m= 100m/s2

5. B B A C C Vectors and 2D vector addition • The sum of two vectors is another vector. A = B + C D = B + 2C ???

6. B B - C -C C B Different direction and magnitude ! A 2D Vector subtraction • Vector subtraction can be defined in terms of addition. = B + (-1)C B - C A = B + C

7. U = |U| û û Reference vectors: Unit Vectors • A Unit Vector is a vector having length 1 and no units • It is used to specify a direction. • Unit vector u points in the direction of U • Often denoted with a “hat”: u = û • Useful examples are the cartesian unit vectors [i, j, k] • Point in the direction of the x, yand z axes. R = rx i + ry j + rz k y j x i k z

8. By C B Bx A Ay Ax Vector addition using components: • Consider, in 2D, C=A + B. (a) C = (Ax i + Ayj ) + (Bxi + Byj ) = (Ax + Bx )i + (Ay + By ) (b)C = (Cx i+ Cy j ) • Comparing components of (a) and (b): • Cx = Ax + Bx • Cy = Ay + By • |C| =[(Cx)2+ (Cy)2 ]1/2

9. ExampleVector Addition • {3,-4,2} • {4,-2,5} • {5,-2,4} • None of the above • Vector A = {0,2,1} • Vector B = {3,0,2} • Vector C = {1,-4,2} What is the resultant vector, D, from adding A+B+C?

10. ExampleVector Addition • {3,-4,2} • {4,-2,5} • {5,-2,4} • None of the above • Vector A = {0,2,1} • Vector B = {3,0,2} • Vector C = {1,-4,2} What is the resultant vector, D, from adding A+B+C?

11. tan-1 ( y / x ) Converting Coordinate Systems • In polar coordinates the vector R= (r,q) • In Cartesian the vectorR= (rx,ry) = (x,y) • We can convert between the two as follows: y (x,y) r ry  rx x • In 3D cylindrical coordinates (r,q,z), r is the same as the magnitude of the vector in the x-y plane [sqrt(x2 +y2)]

12. Resolving vectors into componentsA 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 

13. y’ x’ g q q y x Resolving vectors, little g & the inclined plane • g (bold face, vector) can be resolved into its x,yorx’,y’ components • g = - g j • g = - g cos qj’+ g sin qi’ • The bigger the tilt the faster the acceleration….. along the incline

14. Dynamics II: Motion along a line but with a twist(2D dimensional motion, magnitude and directions) • Particle motions involve a path or trajectory • Recall instantaneous velocity and acceleration • These are vector expressions reflecting x, y & z motion r = r(t) v = dr / dt a = d2r / dt2

15. v Instantaneous Velocity • But how we think about requires knowledge of the path. • The direction of the instantaneous velocity is alonga line that is tangent to the path of the particle’s direction of motion. • The magnitude of the instantaneous velocity vector is the speed, s. (Knight uses v) s = (vx2 + vy2 + vz )1/2

16. Average Acceleration • The average acceleration of particle motion reflects changes in the instantaneous velocity vector (divided by the time interval during which that change occurs). • The average acceleration is a vector quantity directed along ∆v ( a vector! )

17. Instantaneous Acceleration • The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero • The instantaneous acceleration is a vector with components parallel (tangential) and/or perpendicular (radial) to the tangent of the path • 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)

18. Lecture 4 Assignment: Read through Chapter 4 MP Problem Set 2 due Wednesday