1 chapter 2 motion in one dimension 2 3 2.1 kinematics describes motion while ignoring the agents...
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Chapter 2
Motion in One Dimension
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2.1 Kinematics Describes motion while ignoring the
agents that caused the motion For now, will consider motion in one
dimension Along a straight line
Will use the particle model A particle is a point-like object, has mass
but infinitesimal size
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Position Defined in terms of a
frame of reference One dimensional, so
generally the x- or y-axis
The object’s position is its location with respect to the frame of reference
Fig 2.1(a)
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Position-Time Graph The position-time
graph shows the motion of the particle (car)
The smooth curve is a guess as to what happened between the data points
Fig 2.1(b)
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Displacement Defined as the change in position during
some time interval Represented as x
SI units are meters (m) x can be positive or negative
Different than distance – the length of a path followed by a particle
if xxx
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Vectors and Scalars Vector quantities need both magnitude
(size or numerical value) and direction to completely describe them Will use + and – signs to indicate vector
directions Scalar quantities are completely
described by magnitude only
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Average Velocity The average velocity is rate at which
the displacement occurs
The dimensions are length / time [L/T] The SI units are m/s Is also the slope of the line in the
position – time graph
t
xx
t
xv if
average
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Average Velocity, cont Gives no details about the motion Gives the result of the motion It can be positive or negative
It depends on the sign of the displacement It can be interpreted graphically
It will be the slope of the position-time graph
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Displacement from a Velocity - Time Graph
The displacement of a particle during the time interval ti to tf is equal to the area under the curve between the initial and final points on a velocity-time curve
Fig 2.1(c)
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2.2 Instantaneous Velocity The limit of the average velocity as the
time interval becomes infinitesimally short, or as the time interval approaches zero
The instantaneous velocity indicates what is happening at every point of time
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Fig 2.2(a)
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Instantaneous Velocity, graph The instantaneous
velocity is the slope of the line tangent to the x vs. t curve
This would be the green line
The blue lines show that as t gets smaller, they approach the green line
Fig 2.2(b)
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Active Figure2.2
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Instantaneous Velocity, equations The general equation for instantaneous
velocity is
When “velocity” is used in the text, it will mean the instantaneous velocity
dtdx
tx
v lim0t
x
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Instantaneous Velocity, Signs
At point A, vx is positive The slope is positive
At point B, vx is zero The slope is zero
At point C, vx is negative The slope is negative
Fig 2.3
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Instantaneous Speed The instantaneous speed is the
magnitude of the instantaneous velocity vector Speed can never be negative
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Fig 2.4
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Fig 2.5
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2.3 Constant Velocity If the velocity of a particle is constant
Its instantaneous velocity at any instant is the same as the average velocity over a given time period
vx = v x avg = x / t
Also, xf = xi + vx t
These equations can be applied to particles or objects that can be modeled as a particle moving under constant velocity
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Fig 2.6
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2.4 Average Acceleration Acceleration is the rate of change of the
velocity
It is a measure of how rapidly the velocity is changing
Dimensions are L/T2
SI units are m/s2
tv
ttvv
a x
if
xixfavgx
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Instantaneous Acceleration The instantaneous acceleration is the
limit of the average acceleration as t approaches 0
2
2xx
0tx dtxd
dtdv
tv
lima
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Instantaneous Acceleration – Graph The slope of the velocity
vs. time graph is the acceleration
Positive values correspond to where velocity in the positive x direction is increasing
The acceleration is negative when the velocity is in the positive x direction and is decreasing
Fig 2.7
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Fig 2.8
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Fig 2.9
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Fig 2.10
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2.5 Negative Acceleration Negative acceleration does not
necessarily mean that an object is slowing down If the velocity is negative and the
acceleration is negative, the object is speeding up
Deceleration is commonly used to indicate an object is slowing down, but will not be used in this text
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Acceleration and Velocity, 1 When an object’s velocity and
acceleration are in the same direction, the object is speeding up in that direction
When an object’s velocity and acceleration are in the opposite direction, the object is slowing down
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Acceleration and Velocity, 2
The car is moving with constant positive velocity (shown by red arrows maintaining the same size)
Acceleration equals zero
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Acceleration and Velocity, 3
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) This shows positive acceleration and positive velocity
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Acceleration and Velocity, 4
Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the
same length) Velocity is decreasing (red arrows are getting shorter) Positive velocity and negative acceleration
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Fig 2.11
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2.6 Kinematic Equations – Summary
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Fig 2.12
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Kinematic Equations The kinematic equations may be used
to solve any problem involving one-dimensional motion with a constant acceleration
You may need to use two of the equations to solve one problem
The equations are useful when you can model a situation as a particle under constant acceleration
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Kinematic Equations, specific For constant a, Can determine an object’s velocity at
any time t when we know its initial velocity and its acceleration
Does not give any information about displacement
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Kinematic Equations, specific For constant acceleration,
The average velocity can be expressed as the arithmetic mean of the initial and final velocities
Only valid when acceleration is constant
2vv
v xfxix
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Kinematic Equations, specific For constant acceleration,
This gives you the position of the particle in terms of time and velocities
Doesn’t give you the acceleration
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Kinematic Equations, specific For constant acceleration,
Gives final position in terms of velocity and acceleration
Doesn’t tell you about final velocity
2xxiif ta
21
tvxx
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Kinematic Equations, specific For constant a,
Gives final velocity in terms of acceleration and displacement
Does not give any information about the time
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Graphical Look at Motion – displacement – time curve The slope of the
curve is the velocity The curved line
indicates the velocity is changing Therefore, there is
an acceleration
Fig 2.12(a)
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Graphical Look at Motion – velocity – time curve The slope gives the
acceleration The straight line
indicates a constant acceleration
Fig 2.12(b)
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The zero slope indicates a constant acceleration
Graphical Look at Motion – acceleration – time curve
Fig 2.12(c)
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• If you can't see the image above, please install Shockwave Flash Player.• If this active figure can’t auto-play, please click right button, then click play.
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Active Figure2.12
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Problem-Solving Strategy –Constant Acceleration Conceptualize
Establish a mental representation Categorize
Confirm there is a particle or an object that can be modeled as a particle
Confirm it is moving with a constant acceleration
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Problem-Solving Strategy –Constant Acceleration, cont Analyze
Set up the mathematical representation Choose the instant to call the initial time
and another to be the final time These are defined by the parts of the
problem of interest, not necessarily the actual beginning and ending of the motion
Choose the appropriate kinematic equation(s)
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Problem-Solving Strategy –Constant Acceleration, Final Finalize
Check to see if the answers are consistent with the mental and pictorial representations
Check to see if your results are realistic
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2.7 Freely Falling Objects A freely falling object is any object
moving freely under the influence of gravity alone
It does not depend upon the initial motion of the object Dropped – released from rest Thrown downward Thrown upward
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Galileo Galilei(1564-1642)
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Acceleration of Freely Falling Object The acceleration of an object in free fall is
directed downward, regardless of the initial motion
The magnitude of free fall acceleration is g = 9.80 m/s2
g decreases with increasing altitude g varies with latitude 9.80 m/s2 is the average at the earth’s surface
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Acceleration of Free Fall, cont. We will neglect air resistance Free fall motion is constantly
accelerated motion in one dimension Let upward be positive Use the kinematic equations with ay = g
= -9.80 m/s2
The negative sign indicates the direction of the acceleration is downward
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Fig 2.14
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Fig 2.15
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Free Fall Example Initial velocity at A is upward (+) and
acceleration is g (-9.8 m/s2) At B, the velocity is 0 and the
acceleration is g (-9.8 m/s2) At C, the velocity has the same
magnitude as at A, but is in the opposite direction
The displacement is –50.0 m (it ends up 50.0 m below its starting point)
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2.8
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