Physics 207, Lecture 3

1. Physics 207, Lecture 3 • Today (Finish Ch. 2 & start Ch. 3) • Understand acceleration in systems with 1-dimensional motion and non-zero acceleration (usually constant) • Solve problems with zero and constant acceleration (including free-fall and motion on an incline) • Use Cartesian and polar coordinate systems • Perform vector algebra Reading Assignment: For Wednesday: Read Chapter 3 (carefully) through 4.4

2. “2D” Position, Displacement y time (sec) 1 2 3 4 5 6 position -2,2 -1,2 0,2 1,2 2,2 3,2 (x,y meters) x position vectors origin

3. x displacement vectors “2D” Position, Displacement y time (sec) 1 2 3 4 5 6 position -2,2 -1,2 0,2 1,2 2,2 3,2 (x,y meters) position vectors origin

4. x displacement vectors 12 23 34 45 56 Position, Displacement, Velocity y time (sec) 1 2 3 4 5 6 velocity vectors Velocity always has same magnitude & length  CONSTANT

5. v0 v1 v1 v1 v3 v5 v2 v4 v0 Acceleration • Particle motion often involves non-zero acceleration • The magnitude of the velocity vector may change • The direction of the velocity vector may change (true even if the magnitude remains constant) • Both may change simultaneously • E.g., a “particle” with smoothly decreasing speed

6. Average & Instantaneous Acceleration • Average acceleration

7. Average & Instantaneous Acceleration • Average acceleration • The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero

8. Average & Instantaneous Acceleration • Average acceleration • The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero • Note: Position, velocity & acceleration are all vectors, they cannot be added directly to one another (different dimensional units)

9. x t vx t Position, velocity & acceleration for motion along a line • If the positionx is known as a function of time, then we can find both the instantaneous velocityvxand instantaneous accelerationax as a function of time! ax t

10. x t vx t Position, velocity & acceleration for motion along a line • If the positionx is known as a function of time, then we can find both the instantaneous velocityvxand instantaneous accelerationax as a function of time! ax t

11. v1 v0 v3 v5 v2 v4 a a a a a a t 0 Going the other way…. • Particle motion with constant acceleration • The magnitude of the velocity vector changes • A particle with smoothly decreasing speed: v a Dt vf = vi + a Dt = vi + a (tf - ti ) t 0 tf ti a Dt = area under curve = Dv (an integral)

12. So if constant acceleration we can integrate to get explicit v and a x x0 t vx v0 t ax t

13. Rearranging terms gives two other relationships • If constant acceleration then we also get: Slope of x(t) curve

14. vx t An example problem • A particle moves to the right first for 2 seconds at 1 m/s and then 4 seconds at 2 m/s. • What was the average velocity? • Two legs with constant velocity but …. Slope of x(t) curve

15. vx t An example problem • A particle moves to the right first for 2 seconds at 1 m/s and then 4 seconds at 2 m/s. • What was the average velocity? • Two legs with constant velocity but …. • We must find the displacement (x2 –x0) • And x1 = x0 + v0 (t1-t0) x2 = x1 + v1 (t2-t1) • Displacement is (x2 - x1) + (x1 – x0) = v1 (t2-t1) + v0 (t1-t0) • x2 –x0 = 1 m/s (2 s) + 2 m/s (4 s) = 10 m in 6 seconds or 5/3 m/s Slope of x(t) curve

16. A particle starting at rest & moving along a line with constant acceleration has a displacement whose magnitude is proportional to t2 1. This can be tested2. This is a potentially useful result

17. Speed can’t really kill but acceleration may… “High speed motion picture camera frame: John Stapp is caught in the teeth of a massive deceleration. One might have expected that a test pilot or an astronaut candidate would be riding the sled; instead there was Stapp, a mild mannered physician and diligent physicist with a notable sense of humor. Source: US Air Force photo

18. Free Fall • When any object is let go it falls toward the ground !! The force that causes the objects to fall is called gravity. • This acceleration on the Earth’s surface, caused by gravity, is typically written as “little” g • Any object, be it a baseball or an elephant, experiences the same acceleration (g) when it is dropped, thrown, spit, or hurled, i.e. g is a constant.

19. A “biology” experiment • Hypothesis: Older people have slower reaction times • Distance accentuates the impact of time differences • Equipment: Ruler and four volunteers • Older student • Younger student • Record keeper • Statistician • Expt. require multiple trials to reduce statistical errors.

20. Gravity facts: • g does not depend on the nature of the material ! • Galileo (1564-1642) figured this out without fancy clocks & rulers! • Feather & penny behave just the same in vacuum • Nominally, g= 9.81 m/s2 • At the equator g = 9.78 m/s2 • At the North pole g = 9.83 m/s2

21. Gravity Map of the Earth (relief exaggerated) A person off the coast of India would weigh 1% less than at most other places on earth.

22. Gravity map of the US Red: Areas of stronger local g Blue: Areas of weaker local g Due to density variations of the Earth’s crust and mantle

23. y When throwing a ball straight up, which of the following is true about its velocity v and its acceleration aat the highest point in its path? Exercise 1Motion in One Dimension • Bothv = 0anda = 0 • v  0, but a = 0 • v = 0, but a  0 • None of the above

24. When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path? Exercise 1Motion in One Dimension • Bothv = 0anda = 0 • v  0, but a = 0 • v = 0, but a  0 • None of the above y

25. t a v v v t Exercise 2 More complex Position vs. Time Graphs In driving from Madison to Chicago, initially my speed is at a constant 65 mph. After some time, I see an accident ahead of me on I-90 and must stop quickly so I decelerate increasingly “fast” until I stop. The magnitude of myacceleration vs timeis given by, •  •   •   • Question: My velocity vs time graph looks like which of the following ?

26. Exercise 3 1D Freefall • Alice and Bill are standing at the top of a cliff of heightH. Both throw a ball with initial speedv0, Alice straightdownand Bill straightup. • The speed of the balls when they hit the ground arevAandvBrespectively. (Neglect air resistance.) • vA < vB • vA= vB • vA > vB Alice v0 Bill v0 H vA vB

27. Exercise 3 1D Freefall : Graphical solution • Alice and Bill are standing at the top of a cliff of heightH. Both throw a ball with initial speedv0, Alice straightdownand Bill straightup. back at cliff cliff Dv= -g Dt turnaround point v0 vx t identical displacements (one + and one -) -v0 vground ground ground

28. The graph at right shows the y velocity versus time graph for a ball. Gravity is acting downward in the -y direction and the x-axis is along the horizontal. Which explanation best fits the motion of the ball as shown by the velocity-time graph below? Home Exercise,1D Freefall • The ball is falling straight down, is caught, and is then thrown straight down with greater velocity. • The ball is rolling horizontally, stops, and then continues rolling. • The ball is rising straight up, hits the ceiling, bounces, and then falls straight down. • The ball is falling straight down, hits the floor, and then bounces straight up. • The ball is rising straight up, is caught and held for awhile, and then is thrown straight down.

29. Problem Solution Method: Five Steps: • Focus the Problem - draw a picture – what are we asking for? • Describe the physics • what physics ideas are applicable • what are the relevant variables known and unknown • Plan the solution • what are the relevant physics equations • Execute the plan • solve in terms of variables • solve in terms of numbers • Evaluate the answer • are the dimensions and units correct? • do the numbers make sense?

30. Coordinate Systems and vectors • In 1 dimension, only 1 kind of system, • Linear Coordinates (x) +/- • In 2 dimensions there are two commonly used systems, • Cartesian Coordinates (x,y) • Circular Coordinates (r,q) • In 3 dimensions there are three commonly used systems, • Cartesian Coordinates (x,y,z) • Cylindrical Coordinates (r,q,z) • Spherical Coordinates (r,q,f)

31. Vectors • In 1 dimension, we can specify direction with a + or - sign. • In 2 or 3 dimensions, we need more than a sign to specify the direction of something: • To illustrate this, consider the position vectorr in 2 dimensions. Example: Where is Boston ? • Choose origin at New York • Choose coordinate system Boston is 212 milesnortheastof New York [ in (r,q) ]OR Boston is 150 milesnorth and 150 mileseast of New York [ in (x,y) ] Boston r New York

32. A A Vectors... • There are two common ways of indicating that something is a vector quantity: • Boldface notation: A • “Arrow” notation: A =

33. A A = C B C A = B, B = C Vectors have rigorous definitions • A vector is composed of a magnitude and a direction • Examples: displacement, velocity, acceleration • Magnitude of A is designated |A| • Usually vectors include units (m, m/s, m/s2) • For vector algebra a vector has no particular position (Note: the position vector reflects displacement from the origin) • Two vectors are equal if their directions, magnitudes and units match.

34. Comparing Vectors and Scalars • A scalar is an ordinary number. • A magnitude 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 • The product of a vector and a scalar is another vector in the same “direction” but with modified magnitude. B A = -0.75 B A

35. Vector addition • The sum of two vectors is another vector. A = B + C B B A C C

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

37. See you Wednesday Assignment: • For Wednesday, Read through Chapter 4.4 • Mastering Physics Problem Set 2

38. U = |U| û 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 A special vector: Unit Vector • A Unit Vector is defined as 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

39. 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 y’ x’ g q q

40. By C B Bx A Ay Ax Vector addition using components: • Consider C=A + B. (a) C = (Ax i + Ayj ) + (Bxi + Byj ) = (Ax + Bx )i + (Ay + By )j (b)C = (Cx i+ Cy j ) • Comparing components of (a) and (b): • Cx = Ax + Bx • Cy = Ay + By

41. Exercise Vector Addition • {3,-4,2} • {4,-2,5} • {5,-2,4} • 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?

42. 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)]