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Chapter 2 Lecture

Chapter 2 Lecture. Chapter 2 Kinematics in One Dimension. Chapter Goal: To learn how to solve problems about motion in a straight line. Slide 2-2. Chapter 2 Preview. Kinematics is the name for the mathematical description of motion.

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Chapter 2 Lecture

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  1. Chapter 2 Lecture

  2. Chapter 2 Kinematics in One Dimension Chapter Goal: To learn how to solve problems about motion in a straight line. Slide 2-2

  3. Chapter 2 Preview • Kinematics is the name for the mathematical description of motion. • This chapter deals with motion along a straight line, i.e., runners, rockets, skiers. • The motion of an object is described by its position, velocity, and acceleration. • In one dimension, these quantities are represented by x,vx,and ax. • You learned to show these on motion diagrams in Chapter 1. Slide 2-3

  4. Chapter 2 Preview Slide 2-4

  5. Chapter 2 Preview Slide 2-5

  6. Chapter 2 Reading Quiz Slide 2-6

  7. Reading Question 2.1 The slope at a point on a position-versus-time graph of an object is • The object’s speed at that point. • The object’s average velocity at that point. • The object’s instantaneous velocity at that point. • The object’s acceleration at that point. • The distance traveled by the object to that point. Slide 2-7

  8. Reading Question 2.1 The slope at a point on a position-versus-time graph of an object is • The object’s speed at that point. • The object’s average velocity at that point. • The object’s instantaneous velocity at that point. • The object’s acceleration at that point. • The distance traveled by the object to that point. Slide 2-8

  9. Reading Question 2.2 The area under a velocity-versus-time graph of an object is • The object’s speed at that point. • The object’s acceleration at that point. • The distance traveled by the object. • The displacement of the object. • This topic was not covered in this chapter. Slide 2-9

  10. Reading Question 2.2 The area under a velocity-versus-time graph of an object is • The object’s speed at that point. • The object’s acceleration at that point. • The distance traveled by the object. • The displacement of the object. • This topic was not covered in this chapter. Slide 2-10

  11. Reading Question 2.3 The slope at a point on a velocity-versus-time graph of an object is • The object’s speed at that point. • The object’s instantaneous acceleration at that point. • The distance traveled by the object. • The displacement of the object. • The object’s instantaneous velocity at that point. Slide 2-11

  12. Reading Question 2.3 The slope at a point on a velocity-versus-time graph of an object is • The object’s speed at that point. • The object’s instantaneous acceleration at that point. • The distance traveled by the object. • The displacement of the object. • The object’s instantaneous velocity at that point. Slide 2-12

  13. Reading Question 2.4 Suppose we define the y-axis to point vertically upward. When an object is in free fall, it has acceleration in the y-direction • ayg, where g9.80 m/s2. • ayg, where g9.80 m/s2. • Which is negative and increases in magnitude as it falls. • Which is negative and decreases in magnitude as it falls. • Which depends on the mass of the object. Slide 2-13

  14. Reading Question 2.4 Suppose we define the y-axis to point vertically upward. When an object is in free fall, it has acceleration in the y-direction • ayg, where g9.80 m/s2. • ayg, where g9.80 m/s2. • Which is negative and increases in magnitude as it falls. • Which is negative and decreases in magnitude as it falls. • Which depends on the mass of the object. Slide 2-14

  15. Reading Question 2.5 At the turning point of an object, • The instantaneous velocity is zero. • The acceleration is zero. • Both A and B are true. • Neither A nor B is true. • This topic was not covered in this chapter. Slide 2-15

  16. Reading Question 2.5 At the turning point of an object, • The instantaneous velocity is zero. • The acceleration is zero. • Both A and B are true. • Neither A nor B is true. • This topic was not covered in this chapter. Slide 2-16

  17. Reading Question 2.6 A 1-pound block and a 100-pound block are placed side by side at the top of a frictionless hill. Each is given a very light tap to begin their race to the bottom of the hill. In the absence of air resistance • The 1-pound block wins the race. • The 100-pound block wins the race. • The two blocks end in a tie. • There’s not enough information to determine which block wins the race. Slide 2-17

  18. Reading Question 2.6 A 1-pound block and a 100-pound block are placed side by side at the top of a frictionless hill. Each is given a very light tap to begin their race to the bottom of the hill. In the absence of air resistance • The 1-pound block wins the race. • The 100-pound block wins the race. • The two blocks end in a tie. • There’s not enough information to determine which block wins the race. Slide 2-18

  19. Chapter 2 Content, Examples, and QuickCheck Questions Slide 2-19

  20. Uniform Motion • If you drive your car at a perfectly steady 60 mph, this means you change your position by 60 miles for every time interval of 1 hour. • Uniform motion is when equal displacements occur during any successive equal-time intervals. • Uniform motion is always along a straight line. Riding steadily over level ground is a good example of uniform motion. Slide 2-20

  21. Uniform Motion • An object’s motion is uniform if and only if its position-versus-time graph is a straight line. • The average velocity is the slope of the position-versus-time graph. • The SI units of velocity are m/s. Slide 2-21

  22. Tactics: Interpreting Position-versus-Time Graphs Slide 2-22

  23. Example 2.1 Skating with Constant Velocity Slide 2-23

  24. Example 2.1 Skating with Constant Velocity Slide 2-24

  25. Example 2.1 Skating with Constant Velocity Slide 2-25

  26. Example 2.1 Skating with Constant Velocity Slide 2-26

  27. The Mathematics of Uniform Motion • Consider an object in uniform motion along the s-axis, as shown in the graph. • The object’s initial position is si at timeti. • At a later time tf the object’s final position is sf. • The change in time is ttfti. • The final position can be found as: Slide 2-27

  28. Scalars and Vectors • The distance an object travels is a scalar quantity, independent of direction. • The displacement of an object is a vector quantity, equal to the final position minus the initial position. • An object’s speed v is scalar quantity, independent of direction. • Speed is how fast an object is going; it is always positive. • Velocity is a vector quantity that includes direction. • In one dimension the direction of velocity is specified by the  or  sign. Slide 2-28

  29. QuickCheck 2.1 An ant zig-zags back and forth on a picnic table as shown. The ant’s distance traveled and displacement are • 50 cm and 50 cm. • 30 cm and 50 cm. • 50 cm and 30 cm. • 50 cm and –50 cm. • 50 cm and –30 cm. Slide 2-29

  30. QuickCheck 2.1 An ant zig-zags back and forth on a picnic table as shown. The ant’s distance traveled and displacement are • 50 cm and 50 cm. • 30 cm and 50 cm. • 50 cm and 30 cm. • 50 cm and –50 cm. • 50 cm and –30 cm. Slide 2-30

  31. Instantaneous Velocity • An object that is speeding up or slowing down is not in uniform motion. • In this case, the position-versus-time graph is not a straight line. • We can determine the average speed vavg between any two times separated by time interval t by finding the slope of the straight-line connection between the two points. • The instantaneous velocity is the object’s velocity at a single instant of time t. • The average velocity vavgs/t becomes a better and better approximation to the instantaneous velocity as t gets smaller and smaller. Slide 2-31

  32. Instantaneous Velocity Motion diagrams and position graphs of an accelerating rocket. Slide 2-32

  33. Instantaneous Velocity • As ∆t continues to get smaller, the average velocity vavg ∆s/∆t reaches a constant or limiting value. • The instantaneous velocity at time t is the average velocity during a time interval ∆t centered on t, as ∆t approaches zero. • In calculus, this is called the derivative of s with respect to t. • Graphically, ∆s/∆t is the slope of a straight line. • In the limit ∆t 0, the straight line is tangent to the curve. • The instantaneous velocity at time t is the slope of the line that is tangent to the position-versus-time graph at time t. Slide 2-33

  34. QuickCheck 2.2 The slope at a point on a position-versus-time graph of an object is • The object’s speed at that point. • The object’s velocity at that point. • The object’s acceleration at that point. • The distance traveled by the object to that point. • I really have no idea. Slide 2-34

  35. QuickCheck 2.2 The slope at a point on a position-versus-time graph of an object is • The object’s speed at that point. • The object’s velocity at that point. • The object’s acceleration at that point. • The distance traveled by the object to that point. • I really have no idea. Slide 2-35

  36. Example 2.4 Finding Velocity from Position Graphically Slide 2-36

  37. Example 2.4 Finding Velocity from Position Graphically Slide 2-37

  38. Example 2.4 Finding Velocity from Position Graphically Slide 2-38

  39. Example 2.4 Finding Velocity from Position Graphically Slide 2-39

  40. QuickCheck 2.3 Here is a motion diagram of a car moving along a straight road: Which position-versus-time graph matches this motion diagram? Slide 2-40

  41. QuickCheck 2.3 Here is a motion diagram of a car moving along a straight road: Which position-versus-time graph matches this motion diagram? Slide 2-41

  42. QuickCheck 2.4 Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram? E. None of the above. Slide 2-42

  43. QuickCheck 2.4 Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram? E. None of the above. Slide 2-43

  44. QuickCheck 2.5 Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram? Slide 2-44

  45. QuickCheck 2.5 Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram? Slide 2-45

  46. A Little Calculus: Derivatives • ds/dt is called the derivative of s with respect to t. • ds/dt is the slope of the line that is tangent to the position-versus-time graph. • Consider a function u that depends on time as uctn, where c and n are constants: • The derivative of a constant is zero: • The derivative of a sum is the sum of the derivatives. If u and ware two separate functions of time, then: Slide 2-46

  47. Derivative Example • Velocity is the derivative of s with respect to t: • The figure shows the particle’s position and velocity graphs. • The value of the velocity graph at any instant of time is the slope of the position graph at that same time. Suppose the position of a particle as a function of time is s = 2t2 m where t is in s. What is the particle’s velocity? Slide 2-47

  48. QuickCheck 2.6 Here is a position graph of an object: At t =1.5s, the object’s velocity is • 40 m/s. • 20 m/s. • 10 m/s. • –10 m/s. • None of the above. Slide 2-48

  49. QuickCheck 2.6 Here is a position graph of an object: At t = 1.5 s, the object’s velocity is • 40 m/s. • 20 m/s. • 10 m/s. • –10 m/s. • None of the above. Slide 2-49

  50. QuickCheck 2.7 Here is a position graph of an object: At t = 3.0 s, the object’s velocity is • 40 m/s. • 20 m/s. • 10 m/s. • –10 m/s. • None of the above. Slide 2-50

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