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Chapter Goal: To learn how to solve problems about motion in a straight line.

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Title: Chapter Goal: To learn how to solve problems about motion in a straight line.


1
Chapter 2 Kinematics in One Dimension
  • Chapter Goal To learn how to solve problems
    about motion in a straight line.

Slide 2-2
2
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
3
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
4
Uniform Motion
  • An objects 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
5
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 objects 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
6
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 objects
    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
7
Instantaneous Velocity
Motion diagrams and position graphs of an
accelerating rocket.
Slide 2-32
8
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
9
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 w are two separate
    functions of time, then

Slide 2-46
10
Derivative Example
  • Suppose the position of a particle as a function
    of time is s 2t2 m where t is in s. What is the
    particles velocity?
  • Velocity is the derivative of s with respect to
    t
  • The figure shows the particles 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.

Slide 2-47
11
Finding Position from Velocity
  • Suppose we know an objects position to be si at
    an initial time ti.
  • We also know the velocity as a function of time
    between ti and some later time tf.
  • Even if the velocity is not constant, we can
    divide the motion into N steps in which it is
    approximately constant, and compute the final
    position as
  • The curlicue symbol is called an integral.
  • The expression on the right is read, the
    integral of vs dt from ti to tf.

Slide 2-54
12
Finding Position From Velocity
  • The integral may be interpreted graphically as
    the total area enclosed between the t-axis and
    the velocity curve.
  • The total displacement ?s is called the area
    under the curve.

Slide 2-55
13
Example 2.6 The Displacement During a Drag Race
Slide 2-58
14
A Little More Calculus Integrals
  • Taking the derivative of a function is equivalent
    to finding the slope of a graph of the function.
  • Similarly, evaluating an integral is equivalent
    to finding the area under a graph of the
    function.
  • Consider a function u that depends on time as u ?
    ctn, where c and n are constants
  • The vertical bar in the third step means the
    integral evaluated at tf minus the integral
    evaluated at ti.
  • The integral of a sum is the sum of the
    integrals. If u and w are two separate functions
    of time, then

Slide 2-60
15
Motion with Constant Acceleration
  • The SI units of acceleration are (m/s)/s, or
    m/s2.
  • It is the rate of change of velocity and measures
    how quickly or slowly an objects velocity
    changes.
  • The average acceleration during a time interval
    ?t is
  • Graphically, aavg is the slope of a straight-line
    velocity-versus-time graph.
  • If acceleration is constant, the acceleration as
    is the same as aavg.
  • Acceleration, like velocity, is a vector quantity
    and has both magnitude and direction.

Slide 2-64
16
The Kinematic Equations of Constant Acceleration
  • Suppose we know an objects velocity to be vis at
    an initial time ti.
  • We also know the object has a constant
    acceleration of as over the time interval ?t ? tf
    - ti.
  • We can then find the objects velocity at the
    later time tf as

Slide 2-81
17
The Kinematic Equations of Constant Acceleration
  • Suppose we know an objects position to be si at
    an initial time ti.
  • Its constant acceleration as is shown in graph
    (a).
  • The velocity-versus-time graph is shown in graph
    (b).
  • The final position sf is si plus the area under
    the curve of vs between ti and tf

Slide 2-82
18
The Kinematic Equations of Constant Acceleration
  • Suppose we know an objects velocity to be vis at
    an initial position si.
  • We also know the object has a constant
    acceleration of as while it travels a total
    displacement of ?s ? sf - si.
  • We can then find the objects velocity at the
    final position sf

Slide 2-83
19
The Kinematic Equations of Constant Acceleration
Slide 2-84
20
The Kinematic Equations of Constant Acceleration
Motion with constant velocity and constant
acceleration. These graphs assume si 0, vis gt
0, and (for constant acceleration) as gt 0.
Slide 2-85
21
Free Fall
  • The motion of an object moving under the
    influence of gravity only, and no other forces,
    is called free fall.
  • Two objects dropped from the same height will, if
    air resistance can be neglected, hit the ground
    at the same time and with the same speed.
  • Consequently, any two objects in free fall,
    regardless of their mass, have the same
    acceleration

In the absence of air resistance, any two objects
fall at the same rate and hit the ground at the
same time. The apple and feather seen here are
falling in a vacuum.
Slide 2-94
22
Free Fall
  • Figure (a) shows the motion diagram of an object
    that was released from rest and falls freely.
  • Figure (b) shows the objects velocity graph.
  • The velocity graph is a straight line with a
    slope
  • Where g is a positive number which is equal to
    9.80 m/s2 on the surface of the earth.
  • Other planets have different values of g.

Slide 2-95
23
Example 2.14 Finding the Height of a Leap
Slide 2-98
24
Example 2.14 Finding the Height of a Leap
Slide 2-99
25
Example 2.14 Finding the Height of a Leap
Slide 2-100
26
Example 2.14 Finding the Height of a Leap
Slide 2-101
27
Motion on an Inclined Plane
  • Figure (a) shows the motion diagram of an object
    sliding down a straight, frictionless inclined
    plane.
  • Figure (b) shows the the free-fall acceleration
    the object would have if the incline
    suddenly vanished.
  • This vector can be broken into two pieces
    and .
  • The surface somehow blocks , so the
    one-dimensional acceleration along the incline is
  • The correct sign depends on the direction the
    ramp is tilted.

Slide 2-102
28
Example 2.16 From Track to Graphs
Slide 2-106
29
Example 2.16 From Track to Graphs
Slide 2-107
30
Example 2.16 From Track to Graphs
Slide 2-108
31
Example 2.16 From Track to Graphs
Slide 2-109
32
Example 2.19 Finding Velocity from Acceleration
Slide 2-114
33
Example 2.19 Finding Velocity from Acceleration
Slide 2-115
34
Chapter 2 Summary Slides
Slide 2-116
35
General Principles
Slide 2-117
36
General Principles
Slide 2-118
37
Important Concepts
Slide 2-119
38
Important Concepts
Slide 2-120
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