1. Describing Motion: Kinematics in One Dimension
Chapter 2
Describing Motion: Kinematics in One Dimension

2. Units of Chapter 2 Reference Frames and Displacement Average Velocity
Instantaneous Velocity
Acceleration
Motion at Constant Acceleration
Solving Problems
Falling Objects
Graphical Analysis of Linear Motion

3. 2-1 Reference Frames and Displacement
Any measurement of position, distance, or speed must be made with respect to a reference frame.
For example, if you are sitting on a train and someone walks down the aisle, their speed with respect to the train is a few miles per hour, at most. Their speed with respect to the ground is much higher.

4. 2-1 Reference Frames and Displacement
We make a distinction between distance and displacement.
Displacement (blue line) is how far the object is from its starting point, regardless of how it got there.
Distance traveled (dashed line) is measured along the actual path.

5. 2-1 Reference Frames and Displacement
The displacement is written:
Left:
Displacement is positive.
Right:
Displacement is negative.

6. 2-2 Average Velocity
Speed: how far an object travels in a given time interval
(2-1)
Velocity includes directional information:

7. Example 1 Distance Run by a Jogger
2.2 Speed and Velocity
Example 1 Distance Run by a Jogger
How far does a jogger run in 1.5 hours (5400 s)
if his average speed is 2.22 m/s?

8. Average velocity is the displacement divided by the elapsed time.
2.2 Speed and Velocity
Average velocity is the displacement
divided by the elapsed time.

9. 2.2 Speed and Velocity
Example 2 The World’s Fastest Jet-Engine Car
Andy Green in the car ThrustSSC set a world record of
341.1 m/s in To establish such a record, the driver
makes two runs through the course, one in each direction,
to nullify wind effects. From the data, determine the average
velocity for each run.

10. 2.2 Speed and Velocity

11. 2-3 Instantaneous Velocity
The instantaneous velocity is the average velocity, in the limit as the time interval becomes infinitesimally short.
(2-3)
These graphs show (a) constant velocity and (b) varying velocity.

12. 2-4 Acceleration
Acceleration is the rate of change of velocity.

13. 2-4 Acceleration
Acceleration is a vector, although in one-dimensional motion we only need the sign.
The previous image shows positive acceleration; here is negative acceleration:

14. 2-4 Acceleration
There is a difference between negative acceleration and deceleration:
Negative acceleration is acceleration in the negative direction as defined by the coordinate system.
Deceleration occurs when the acceleration is opposite in direction to the velocity.

15. 2-4 Acceleration
The instantaneous acceleration is the average acceleration, in the limit as the time interval becomes infinitesimally short.
(2-5)

16. 2-5 Motion at Constant Acceleration
The average velocity of an object during a time interval t is
The acceleration, assumed constant, is

17. 2.4 Equations of Kinematics for Constant Acceleration
Five kinematic variables:
1. displacement, x
2. acceleration (constant), a
3. final velocity (at time t), v (or vf)
4. initial velocity, vo (or Vi)
5. elapsed time, t

18. Kinematic Equations a = Δ v = v - vo to = 0 Δ t t - to Therefore:
v = vo at

19. Kinematic Equations Vavg = Δx t Δx = vavgt = vo + v t 2
Δx = ½ (vo + v) t
But v = vo at
Δx = ½ (vo + vo + at) t
Δx = vo t + ½ at2

20. Kinematic Equations Δx = ½ (vo + v) t
We can develop an equation without t
vf = vo at
t = v - vo (sub. into equation above for ΔX) a
v2 = vo aΔx

21. Kinematic Equations x a 2 v at 1 t D + = ÷ ø ö ç è æ
Summary Uniform Acceleration x a 2 v at 1 t o
average D + = ÷ ø ö ç è æ

22. Kinematic Equations x a 2 v at 1 t D + = ÷ ø ö ç è æ
Summary Uniform Acceleration x a 2 v at 1 t o
average D + = ÷ ø ö ç è æ

23. 2-5 Motion at Constant Acceleration
We can also combine these equations so as to eliminate t:
We now have all the equations we need to solve constant-acceleration problems.
(2-10)
(2-11a)
(2-11b)
(2-11c)
(2-11d)

24. 2.4 Equations of Kinematics for Constant Acceleration

25. 2.4 Equations of Kinematics for Constant Acceleration
Example 6 Catapulting a Jet
Find its displacement.

26. 2.4 Equations of Kinematics for Constant Acceleration

27. 2.4 Equations of Kinematics for Constant Acceleration

28. Free Fall
All objects moving under the influence of only gravity are said to be in free fall
All objects falling near the earth’s surface fall with a constant acceleration
Galileo originated our present ideas about free fall from his inclined planes
The acceleration is called the acceleration due to gravity, and indicated by g

29. Acceleration due to Gravity
Symbolized by g
g = 9.8 m/s²
g is always directed downward
toward the center of the earth

30. Free Fall -- an object dropped
Initial velocity is zero
Let up be positive
Use the kinematic equations
Generally use y instead of x since vertical
vo= 0
a = g

31. 2-7 Falling Objects
The acceleration due to gravity at the Earth’s surface is approximately 9.80 m/s2.

32. Free Fall -- an object thrown downward
a = g
Initial velocity  0
With upward being positive, initial velocity will be negative

33. Free Fall -- object thrown upward
Initial velocity is upward, so positive
The instantaneous velocity at the maximum height is zero
a = g everywhere in the motion
g is always downward, negative
v = 0

34. Free Fall Example 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

35. Free Fall Example What is the maximum height the ball reaches?
How long does it take to reach the maximum height?
How long is the ball in the air total?
What is the velocity of the ball just before it hits the ground?

36. Combination Motions A rocket moves straight upward,
starting from rest, with an acceleration of a=29.4m/s2 for 4 seconds when the fuel runs out. It continues to coast for a while.
a) What height does it reach above the launch point?
b) What is its velocity just before it
crashes to the ground?

37. Combination Motions
a) What height does it reach above the launch point?
b) What is its velocity just before it
crashes to the ground?

38. Thrown upward, cont. The motion may be symmetrical
then tup = tdown
then vf = -vo
The motion may not be symmetrical
Break the motion into various parts
generally up and down

39. 2-5 Motion at Constant Acceleration
In addition, as the velocity is increasing at a constant rate, we know that
Combining these last three equations, we find:
(2-8)
(2-9)

40. 2-6 Solving Problems
Read the whole problem and make sure you understand it. Then read it again.
Decide on the objects under study and what the time interval is.
Draw a diagram and choose coordinate axes.
Write down the known (given) quantities, and then the unknown ones that you need to find.
What physics applies here? Plan an approach to a solution.

41. 2-6 Solving Problems
6. Which equations relate the known and unknown quantities? Are they valid in this situation? Solve algebraically for the unknown quantities, and check that your result is sensible (correct dimensions).
7. Calculate the solution and round it to the appropriate number of significant figures.
8. Look at the result – is it reasonable? Does it agree with a rough estimate?
9. Check the units again.

42. 2-7 Falling Objects
Near the surface of the Earth, all objects experience approximately the same acceleration due to gravity.
This is one of the most common examples of motion with constant acceleration.

43. 2-7 Falling Objects
In the absence of air resistance, all objects fall with the same acceleration, although this may be hard to tell by testing in an environment where there is air resistance.

44. 2-8 Graphical Analysis of Linear Motion
This is a graph of x vs. t for an object moving with constant velocity. The velocity is the slope of the x-t curve.

45. 2-8 Graphical Analysis of Linear Motion
On the left we have a graph of velocity vs. time for an object with varying velocity; on the right we have the resulting x vs. t curve. The instantaneous velocity is tangent to the curve at each point.

46. 2-8 Graphical Analysis of Linear Motion
The displacement, x, is the area beneath the v vs. t curve.

47. Summary of Chapter 2
Kinematics is the description of how objects move with respect to a defined reference frame.
Displacement is the change in position of an object.
Average speed is the distance traveled divided by the time it took; average velocity is the displacement divided by the time.
Instantaneous velocity is the limit as the time becomes infinitesimally short.

48. Summary of Chapter 2
Average acceleration is the change in velocity divided by the time.
Instantaneous acceleration is the limit as the time interval becomes infinitesimally small.
The equations of motion for constant acceleration are given in the text; there are four, each one of which requires a different set of quantities.
Objects falling (or having been projected) near the surface of the Earth experience a gravitational acceleration of 9.80 m/s2.

49. Kinematics in One Dimension
Chapter 2
Kinematics in One Dimension

50. Kinematics deals with the concepts that
are needed to describe motion.
Dynamics deals with the effect that forces
have on motion.
Together, kinematics and dynamics form
the branch of physics known as Mechanics.

51. 2.1 Displacement

52. 2.1 Displacement

53. 2.1 Displacement

54. 2.1 Displacement

55. SI units for speed: meters per second (m/s)
2.2 Speed and Velocity
Average speed is the distance traveled divided by the time
required to cover the distance.
SI units for speed: meters per second (m/s)

56. The instantaneous velocity indicates how fast
2.2 Speed and Velocity
The instantaneous velocity indicates how fast
the car moves and the direction of motion at each
instant of time.

57. The notion of acceleration emerges when a change in
velocity is combined with the time during which the
change occurs.

58. DEFINITION OF AVERAGE ACCELERATION

59. 2.3 Acceleration
Example 3 Acceleration and Increasing Velocity
Determine the average acceleration of the plane.

60. 2.3 Acceleration

61. 2.3 Acceleration
Example 3 Acceleration and Decreasing
Velocity

62. 2.3 Acceleration

63. 2.4 Equations of Kinematics for Constant Acceleration
It is customary to dispense with the use of boldface symbols
overdrawn with arrows for the displacement, velocity, and
acceleration vectors. We will, however, continue to convey
the directions with a plus or minus sign.

64. 2.4 Equations of Kinematics for Constant Acceleration
Let the object be at the origin when the clock starts.

65. 2.4 Equations of Kinematics for Constant Acceleration

66. 2.4 Equations of Kinematics for Constant Acceleration

67. 2.4 Equations of Kinematics for Constant Acceleration

68. 2.5 Applications of the Equations of Kinematics
Reasoning Strategy
1. Make a drawing.
2. Decide which directions are to be called positive (+) and
negative (-).
3. Write down the values that are given for any of the five
kinematic variables.
4. Verify that the information contains values for at least three
of the five kinematic variables. Select the appropriate equation.
5. When the motion is divided into segments, remember that
the final velocity of one segment is the initial velocity for the next.
6. Keep in mind that there may be two possible answers to a
kinematics problem.

69. 2.5 Applications of the Equations of Kinematics
Example 8 An Accelerating Spacecraft
A spacecraft is traveling with a velocity of m/s. Suddenly
the retrorockets are fired, and the spacecraft begins to slow down
with an acceleration whose magnitude is 10.0 m/s2. What is
the velocity of the spacecraft when the displacement of the craft
is +215 km, relative to the point where the retrorockets began
firing? x a v vo t m
-10.0 m/s2 ?
+3250 m/s

70. 2.5 Applications of the Equations of Kinematics

71. 2.5 Applications of the Equations of Kinematicsx a v vo t m
-10.0 m/s2 ?
+3250 m/s

72. In the absence of air resistance, it is found that all bodies
2.6 Freely Falling Bodies
In the absence of air resistance, it is found that all bodies
at the same location above the Earth fall vertically with
the same acceleration. If the distance of the fall is small
compared to the radius of the Earth, then the acceleration
remains essentially constant throughout the descent.
This idealized motion is called free-fall and the acceleration
of a freely falling body is called the acceleration due to
gravity.

73. 2.6 Freely Falling Bodies

74. Example 10 A Falling Stone
2.6 Freely Falling Bodies
Example 10 A Falling Stone
A stone is dropped from the top of a tall building. After 3.00s
of free fall, what is the displacement y of the stone?

75. 2.6 Freely Falling Bodies y a v vo t ?
-9.80 m/s2
0 m/s
3.00 s

76. 2.6 Freely Falling Bodies y a v vo t ?
-9.80 m/s2
0 m/s
3.00 s

77. Example 12 How High Does it Go? The referee tosses the coin up
2.6 Freely Falling Bodies
Example 12 How High Does it Go?
The referee tosses the coin up
with an initial speed of 5.00m/s.
In the absence if air resistance,
how high does the coin go above
its point of release?

78. 2.6 Freely Falling Bodies y a v vo t ?
-9.80 m/s2
0 m/s
+5.00 m/s

79. 2.6 Freely Falling Bodies y a v vo t ?
-9.80 m/s2
0 m/s
+5.00 m/s

80. Conceptual Example 14 Acceleration Versus Velocity
2.6 Freely Falling Bodies
Conceptual Example 14 Acceleration Versus Velocity
There are three parts to the motion of the coin. On the way
up, the coin has a vector velocity that is directed upward and
has decreasing magnitude. At the top of its path, the coin
momentarily has zero velocity. On the way down, the coin
has downward-pointing velocity with an increasing magnitude.
In the absence of air resistance, does the acceleration of the
coin, like the velocity, change from one part to another?

81. Conceptual Example 15 Taking Advantage of Symmetry
2.6 Freely Falling Bodies
Conceptual Example 15 Taking Advantage of Symmetry
Does the pellet in part b strike the ground beneath the cliff
with a smaller, greater, or the same speed as the pellet
in part a?

82. 2.7 Graphical Analysis of Velocity and Acceleration

83. 2.7 Graphical Analysis of Velocity and Acceleration

84. 2.7 Graphical Analysis of Velocity and Acceleration

85. 2.7 Graphical Analysis of Velocity and Acceleration