Chapter 2 Kinematics Rectilinear Motion. Displacement, Velocity & Acceleration. Equations describing motion along straight line

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1 Chapter 2 Kinematics Rectilinear Motion Displacement, Velocity & Acceleration Equations describing motion along straight line Objects falling freely under earth s gravity

2 MECHANICS DYNAMICS (Force & Response) STATICS (At rest) KINEMATICS (describes motion)

3 Concepts to study objects in motion Position Displacement (similar to distance) Velocity (similar to speed) Acceleration

4 What is Position in Physics? Frame of reference A choice of coordinate axes with an origin Coordinates for representing starting and ending points of a motion One dimension only (this chapter) (example) a straight line extending to the right and left of the point at the center (origin) and called the x direction

5 Displacement Defined as the change in position x x x f f - final position; i - initial Units -meters (SI) i CAN BE POSITIVE OR NEGATIVE ALONG A GIVEN DIRECTION!!!

6 From A to B x i = 30 m x f = 52 m x = 22 m The displacement is positive, indicating the motion was in the positive x direction From C to F x i = 38 m x f = -53 m x = -91 m The displacement is negative, indicating the motion was in the negative x direction

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8 Some examples of displacement Initial x Final x x Sign of x

9 Displacement is a vector (Need both magnitude (size) and direction to completely describe) For 1 D motion just + or sign (Scalar quantities have just magnitude)

10 IMPORTANT NOTE Motion along one dimension may be either along x or y or z axis direction and can be used to represent horizontal side to side (motion along x axis) or front and back (motion along z axis) or vertical up and down motion (motion along y axis)

11 Displacement not always distance Throw a ball straight up and then catch it at the same point you released it Distance = 2 times height reached by ball The displacement is zero!!!!!! x i x f so x is zero

12 Two dimensions Example East Walk 3 blocks east and then 4 blocks north Total distance =? Displacement =?

13 Speed The average speed of an object is defined as the total distance traveled divided by the total time elapsed Average speed v total distance total time d t SI units meters/second m/s Speed - scalar? Vector? Speed always positive?

14 You drove to Trenton campus 8 miles. You drove for 17 minutes and stopped for 3 mins. at a gas station on your way. What was your : Displacement? Distance covered? Average speed?

15 +8 miles relative to origin at WWC 8 miles 0.4 miles/min or 10.7 m/s Think! What is the displacement for the round trip if you returned back to college?

16 Velocity Average velocity = displacement time elapsed v average x xf x t t t SI units m/s (other units feet/s, cm/s) Velocity can be positive or negative Velocity is a vector! For 1D, just +ve or negative Direction of velocity same as displacement f i i

17 You drove to Trenton campus 8 miles. You drove for 17 minutes and stopped for 3 mins. at a gas station on your way. What was your average velocity?

18 Average velocity = m/s What is the average velocity for the round trip?

19 Speed vs. Velocity average velocity = average speed?

20 Note: In text, bars over physical quantities represent average values. e. g. average velocity, average speed, average acceleration

21 Quick Quiz 2.1 Total distance = Displacement = Average velocity in x direction = Average Speed =

22 200 yards, 0, 0, = 8 yards/sec

23 Chapter 2 Lecture 2 understand changing velocity & acceleration; motion with constant acceleration special case of constant acceleration: free fall

24 Understanding Velocity & Acceleration from Graphs Position(x) vs. time (t) & velocity(v) vs. time (t) graph Object moving with a constant velocity (uniform motion) what is the shape of x t graph? Average velocity = slope of line joining initial & final positions Slope of a line is rise divided by run slope change in vertical axis change in horizontal axis

25 comparing constant & changing velocity equal displacements in equal intervals of time; average velocity is constant all the time; uniform motion average velocity (slope of curve) changes from time to time; change in velocity means acceleration!

26 Instantaneous velocity For non uniform velocity, use instantaneous velocity (called simply v) instead of v avg The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero v lim t 0 x t instantaneous velocity indicates what is happening at every point of time when velocity keeps changing

27 Instantaneous Velocity on a Graph The slope of the line tangent to the position-vs.-time graph is defined to be the instantaneous velocity at that time The instantaneous speed is defined as the magnitude of the instantaneous velocity

28 Acceleration Acceleration change in velocity time elapsed a v vf v t t t This is average acceleration Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Customary) Vector? Yes, in 1D + or - f i i

29 What are v i and v f? Instantaneous values of velocities (initial and final) Class Example 1 A car takes 2 seconds to accelerate from an initial velocity of +10 m/s to a final velocity +20 m/s. What is the average acceleration?

30 ā =+5 m/s 2 acceleration is positive and in the same direction as initial velocity

31 Class Example 2 A plane travelling at +30 m/s on the runway stops in 5 seconds. What is the acceleration?

32 ā = -6 m/s 2 (a is negative and it has a sign opposite to that of initial velocity) TRUE OR FALSE So, to slow down an object you always need to have negative acceleration. To speed up an object, you always need to have positive acceleration. FALSE, FALSE, FALSE, FALSE, FALSE, FALSE

33 Example: Motion Diagram of A Car Please insert active figure 2.12

34 Average Acceleration When the sign of the initial velocity and the acceleration are the same (either positive or negative), then the speed is increasing When the sign of the initial velocity and the acceleration are in the opposite directions, the speed is decreasing

35 Negative Acceleration A negative acceleration does not necessarily mean the object is slowing down If the acceleration and velocity are both negative, the object is speeding up

36 Finding Average Acceleration From v-t graph

37 Relationship Between Acceleration and Velocity Uniform velocity (shown by red arrows maintaining the same size) Acceleration equals zero

38 Relationship Between Velocity and Acceleration Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer) Positive velocity and positive acceleration

39 Relationship Between Velocity and Acceleration Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter) Velocity is positive and acceleration is negative

40 True or False? Quick Quiz 2.2 a) A car must always have an acceleration in the same direction as velocity b) It is possible for a slowing car to have +ve acceleration c) An object with constant nonzero acceleration can stop but then can not remain at rest Answers: a. Think about class example 2 b. Think about slowing car with initial velocity of -10 m/s and final velocity -5 m/s c. Think about +ve velocity, -ve acceleration of chalk thrown vertically upward

41 1D motion with const. acceleration Equations with constant acceleration, a v v at o 1 x vt v o v t x vot at v v 2a x o Objects thrown up or down on earth move with a special constant acceleration called g the acceleration due to gravity = 9.8 m/s 2

42 Notes on the equations If initial t=0 & final t=t, then t = t x v av erage t v o 2 v f t Gives displacement as a function of velocity and time Use when you don t know and aren t asked for the acceleration

43 Notes on the equations v v at o Shows velocity as a function of acceleration and time Use when you don t know and aren t asked to find the displacement

44 Notes on the equations x v o t at Gives displacement as a function of time, velocity and acceleration Use when you don t know and aren t asked to find the final velocity 1 2 2

45 Notes on the equations 2 2 o v v 2a x Gives velocity as a function of acceleration and displacement Use when you don t know and aren t asked for the time

46 Textbook Example 2.4, page 37, Daytona 500 Starts from rest: at t=0, v is zero (v 0 = 0) Constant acceleration: uniform a = 5m/s 2 Velocity of car after a distance of 100 ft =? Use equation: v 2 2 2a x v0 For distance convert feet to m! (1 m = feet) Time elapsed=? Use equation: v v 0 at

47 Textbook example 2.4 Ans: 17.5 m/s & 3.50 s End of Chapter MC1: (!!! sig fig & units!!!) Choose upward vertical direction as +ve y axis (coming up: free fall, use y axis) Initial velocity +15 Final velocity -8 (Assume a = -9.8.coming up) Find t Use v v 0 at

48 2.35 s Problem 9 Instantaneous velocities of tennis player at a) 0.50 s b) 2.0 s c) 3.0 s d) 4.5 s Just find slope of graph around each t value

49 4, -4, 0, 2 with sig fig & units

50 Galileo Galilei Galileo formulated the laws that govern the motion of objects in free fall Also looked at: Inclined planes Relative motion Thermometers Pendulum

51 Free Fall All objects moving under the influence of gravity only are said to be in free fall Free fall does not depend on the object s original motion All objects falling near the earth s surface fall with a constant acceleration The acceleration is called the acceleration due to gravity, and indicated by g

52 Acceleration due to Gravity Symbolized by g g = 9.80 m/s² pointing towards the earth g 10 m/s 2 g is always directed downward Toward the center of the earth Ignoring air resistance and assuming g doesn t vary with altitude over short vertical distances, free fall is constantly accelerated motion Who is in free fall? Space shuttle!!!

53 Free Fall an object dropped Initial velocity is zero Let up be positive Use the kinematic equations Tip: Generally use y instead of x since vertical Acceleration is g = m/s 2 v o = 0 a = g

54 Free Fall an object thrown downward a = g = m/s 2 Initial velocity 0 With upward being positive, initial velocity will be negative

55 Free Fall -- object thrown upward Initial velocity is upward, so positive The instantaneous velocity at the maximum height is zero a = g = m/s 2 everywhere in the motion v = 0

56 Thrown upward, cont. The motion may be symmetrical Then t up = t down Then v = -v o The motion may not be symmetrical Break the motion into various parts Generally up and down

57 Problem 45: thrown upward with speed 25 This is free fall motion with uniform a=g=9.8 Upward direction is +ve y axis & use g=-9.8 a) Maximum height reached At this value of x, final v = 0; initial v = 25 Use v 2 2 2a y v0 b) Use v v 0 at

58 31.9m, 2.55s, same t, same initial v & opposite

59 Class Example Obama throws a rock down with speed of 12 m/s from the top of a tower. The rock hits the ground after 2.0s. What is the height of the tower? Choose down as positive & choose y for displacement Given initial velocity (v 0 =12) Given time (t = 2) Given free fall (moving under constant downward acceleration (a=g=9.8) Asked to find y - the distance traveled by rock from top to bottom (height of tower)

60 Use 2 1 y v t a 0 2 t

61 44 m (rounded) In free fall motion of chalk, there is constant acceleration called g of magnitude 9.8 meter per second squared throughout its trip from the time it leaves my hands until it hits the ground

62 Non-symmetrical Free Fall Need to divide the motion into segments Possibilities include Upward and downward portions The symmetrical portion back to the release point and then the non-symmetrical portion