A Mathematical Model of Motion

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A Mathematical Model of Motion

• CHAPTER 5
• PHYSICS

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5.1 Graphing Motion in One Dimension

• Interpret graphs of position versus time for a
  moving object to determine the velocity of the
  object
• Describe in words the information presented in
  graphs and draw graphs from descriptions of
  motion
• Write equations that describe the position of an
  object moving at constant velocity

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Parts of a Graph

• X-axis
• Y-axis
• All axes must be labeled with appropriate units,
  and values.

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5.1 Position vs. Time

• The x-axis is always time
• The y-axis is always position
• The slope of the line indicates the velocity of
  the object.
• Slope (y2-y1)/(x2-x1)
• d1-d0/t1-t0
• ?d/?t

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Steepness of slope on Position-Time graph

• Slope is related to velocity
• Steep slope higher velocity
• Shallow slope less velocity

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Uniform Motion

• Uniform motion is defined as equal displacements
  occurring during successive equal time periods
  (sometimes called constant velocity)
• Straight lines on position-time graphs mean
  uniform motion.

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Given below is a diagram of a ball rolling along
a table. Strobe pictures reveal the position of
the object at regular intervals of time, in this
case, once each 0.1 seconds.  
Notice that the ball covers an equal distance
between flashes. Let's assume this distance
equals 20 cm and display the ball's behavior on a
graph plotting its x-position versus time.
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The slope of this line would equal 20 cm divided
by 0.1 sec or 200 cm/sec. This represents the
ball's average velocity as it moves across the
table. Since the ball is moving in a positive
direction its velocity is positive. That is, the
ball's velocity is a vector quantity possessing
both magnitude (200 cm/sec) and direction
(positive).
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Different Position. Vs. Time graphs
Accelerated Motion
Uniform Motion
Constant positive velocity (zero acceleration)
Increasing positive velocity (positive
acceleration)
Constant negative velocity (zero acceleration)
Decreasing negative velocity (positive
acceleration)
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Different Position. Vs. Time
Changing slope means changing velocity!!!!!!
Increasing negative slope ??
Decreasing negative slope ??
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A Starts at home (origin) and goes forward
slowly B Not moving (position remains constant
as time progresses) C Turns around and goes in
the other direction quickly, passing up home
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During which intervals was he traveling in a
positive direction? During which intervals was he
traveling in a negative direction? During which
interval was he resting in a negative
location? During which interval was he resting in
a positive location? During which two intervals
did he travel at the same speed? A) 0 to 2 sec B)
2 to 5 sec C) 5 to 6 sec D)6 to 7 sec E) 7 to
9 sec F)9 to 11 sec
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Graphing w/ Acceleration
x
A Start from rest south of home increase speed
gradually B Pass home gradually slow to a stop
(still moving north) C Turn around gradually
speed back up again heading south D Continue
heading south gradually slow to a stop near the
starting point
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You try it..

• Using the Position-time graph given to you, write
  a one or two paragraph story that describes the
  motion illustrated.
• You need to describe the specific motion for each
  of the steps (a-f)
• You will be graded upon your ability to correctly
  describe the motion for each step.

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Tangent Lines
x
t
On a position vs. time graph
SLOPE VELOCITY
Positive Positive
Negative Negative
Zero Zero
SLOPE SPEED
Steep Fast
Gentle Slow
Flat Zero
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Increasing Decreasing
Increasing
Decreasing
On a position vs. time graph Increasing means
moving forward (positive direction). Decreasing
means moving backwards (negative direction).
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Concavity
On a position vs. time graph Concave up means
positive acceleration. Concave down means
negative acceleration.
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Special Points
Q
R
P
S
Inflection Pt. P, R Change of concavity, change of acceleration
Peak or Valley Q Turning point, change of positive velocity to negative
Time Axis Intercept P, S Times when you are at home, or at origin
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5.2 Graphing Velocity in One Dimension

• Determine, from a graph of velocity versus time,
  the velocity of an object at a specific time
• Interpret a v-t graph to find the time at which
  an object has a specific velocity
• Calculate the displacement of an object from the
  area under a v-t graph

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5.2 Velocity vs. Time

• X-axis is the time
• Y-axis is the velocity
• Slope of the line the acceleration

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Different Velocity-time graphs
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Different Velocity-time graphs
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Velocity vs. Time

• Horizontal lines constant velocity
• Sloped line changing velocity
• Steeper greater change in velocity per second
• Negative slope deceleration

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Acceleration vs. Time

• Time is on the x-axis
• Acceleration is on the y-axis
• Shows how acceleration changes over a period of
  time.
• Often a horizontal line.

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All 3 Graphs
v
t
a
t
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Real life
Note how the v graph is pointy and the a
graph skips. In real life, the blue points would
be smooth curves and the orange segments would be
connected. In our class, however, well only
deal with constant acceleration.
v
t
a
t
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Constant Rightward Velocity
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Constant Leftward Velocity
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Constant Rightward Acceleration
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Constant Leftward Acceleration
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Leftward Velocity with Rightward Acceleration
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Graph Practice
Try making all three graphs for the following
scenario 1. Newberry starts out north of home.
At time zero hes driving a cement mixer south
very fast at a constant speed. 2. He
accidentally runs over an innocent moose crossing
the road, so he slows to a stop to check on the
poor moose. 3. He pauses for a while until he
determines the moose is squashed flat and deader
than a doornail. 4. Fleeing the scene of the
crime, Newberry takes off again in the same
direction, speeding up quickly. 5. When his
conscience gets the better of him, he slows,
turns around, and returns to the crash site.
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Area Underneath v-t Graph

• If you calculate the area underneath a v-t graph,
  you would multiply height X width.
• Because height is actually velocity and width is
  actually time, area underneath the graph is equal
  to
• Velocity X time or
• Vt

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• Remember that Velocity ?d
• ?t
• Rearranging, we get ?d velocity X ?t
• So.the area underneath a velocity-time graph is
  equal to the displacement during that time
  period.

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Area
Note that, here, the areas are about equal, so
even though a significant distance may have been
covered, the displacement is about zero, meaning
the stopping point was near the starting point.
The position graph shows this as well.
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Velocity vs. Time

• The area under a velocity time graph indicates
  the displacement during that time period.
• Remember that the slope of the line indicates the
  acceleration.
• The smaller the time units the more
  instantaneous is the acceleration at that
  particular time.
• If velocity is not uniform, or is changing, the
  acceleration will be changing, and cannot be
  determined for an instant, so you can only find
  average acceleration

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5.3 Acceleration

• Determine from the curves on a velocity-time
  graph both the constant and instantaneous
  acceleration
• Determine the sign of acceleration using a v-t
  graph and a motion diagram
• Calculate the velocity and the displacement of an
  object undergoing constant acceleration

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5.3 Acceleration

• Like speed or velocity, acceleration is a rate of
  change, defined as the rate of change of velocity
• Average Acceleration change in velocity

Elapsed time
Units of acceleration?
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Rearrangement of the equation
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Finally

• This equation is to be used to find (final)
  velocity of an accelerating object. You can use
  it if there is or is not a beginning velocity

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Displacement under Constant Acceleration

• Remember that displacement under constant
  velocity was

?d vt or d1 d0 vt

• With acceleration, there is no one single
  instantaneous v to use, but we could use an
  average velocity of an accelerating object.

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Average velocity of an accelerating object would
simply be ½ of sum of beginning and ending
velocities
Average velocity of an accelerating object
V ½ (v0 v1)
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So.
Key equation
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Some other equations
2
This equation is to be used to find final
position when there is an initial velocity, but
velocity at time t1 is not known.
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If no time is known, use this to find final
position.
2
2
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Finding final velocity if no time is known
2
2
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The equations of importance
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2
2
2
2
2
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Practical Application Velocity/Position/Time
equations

• Calculation of arrival times/schedules of
  aircraft/trains (including vectors)
• GPS technology (arrival time of signal/distance
  to satellite)
• Military targeting/delivery
• Calculation of Mass movement (glaciers/faults)
• Ultrasound (speed of sound) (babies/concrete/metal
  s) Sonar (Sound Navigation and Ranging)
• Auto accident reconstruction
• Explosives (rate of burn/expansion rates/timing
  with det. cord)

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5.4 Free Fall

• Recognize the meaning of the acceleration due to
  gravity
• Define the magnitude of the acceleration due to
  gravity as a positive quantity and determine the
  sign of the acceleration relative to the chosen
  coordinate system
• Use the motion equations to solve problems
  involving freely falling objects

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Freefall

• Defined as the motion of an object if the only
  force acting on it is gravity.
• No friction, no air resistance, no drag

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Acceleration Due to Gravity

• Galileo Galilei recognized about 400 years ago
  that, to understand the motion of falling
  objects, the effects of air or water would have
  to be ignored.
• As a result, we will investigate falling, but
  only as a result of one force, gravity.

Galileo Galilei 1564-1642
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Galileos Ramps

• Because gravity causes objects to move very fast,
  and because the time-keepers available to Galileo
  were limited, Galileo used ramps with moveable
  bells to slow down falling objects for accurate
  timing.

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Galileos Ramps
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Galileos Ramps

• To keep accurate time, Galileo used a water
  clock.
• For the measurement of time, he employed a large
  vessel of water placed in an elevated position
  to the bottom of this vessel was soldered a pipe
  of small diameter giving a thin jet of water,
  which he collected in a small glass during the
  time of each descent... the water thus collected
  was weighed, after each descent, on a very
  accurate balance the difference and ratios of
  these weights gave us the differences and ratios
  of the times...

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Displacements of Falling Objects

• Looking at his results, Galileo realized that a
  falling ( or rolling downhill) object would have
  displacements that increased as a function of the
  square of the time, or t2
• Another way to look at it, the velocity of an
  object increased as a function of the square of
  time, multiplied by some constant.

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• Galileo also found that all objects, no matter
  what slope of ramp he rolled them down, and as
  long as the ramps were all the same height, would
  reach the bottom with the same velocity.

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Galileos Finding

• Galileo found that, neglecting friction, all
  freely falling objects have the same
  acceleration.

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Hippo Ping Pong Ball
In a vacuum, all bodies fall at the same rate.
If a hippo and a ping pong ball were dropped from
a helicopter in a vacuum (assuming the copter
could fly without air), theyd land at the same
time.
When theres no air resistance, size and shape
matter not!
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Proving Galileo Correct
Galileo could not produce a vacuum to prove his
ideas. That came later with the invention of a
vacuum machine, and the demonstration with a
guinea feather and gold coin dropped in a vacuum.
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Guinea Feather and Coin/NASA demonstrations
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Acceleration Due to Gravity

• Galileo calculated that all freely falling
  objects accelerate at a rate of

9.8 m/s2
This value, as an acceleration, is known as g
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Acceleration Due to Gravity

• Because this value is an acceleration value, it
  can be used to calculate displacements or
  velocities using the acceleration equations
  learned earlier. Just substitute g for the a

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Example problem

• A brick is dropped from a high building.
• What is its velocity after 4.0 sec.?
• How far does the brick fall during this time?

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The Churchs opposition to new thought

• Church leaders of the time held the same views as
  Aristotle, the great philosopher.
• Aristotle thought that objects of different mass
  would fall at different ratesmakes sense
  huh??????
• All objects have their natural position. Rocks
  fall faster than feathers, so it only made sense
  (to him)
• Galileo, in attempting to convince church leaders
  that the natural position view was incorrect,
  considered two rocks of different mass.

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Falling Rock Conundrum

• Galileo presented this in his book Dialogue
  Concerning the Two Chief World Systems(1632) as a
  discussion between Simplicio (as played by a
  church leader) and Salviati (as played by
  Galileo)
• Two rocks of different masses are dropped
• Massive rock falls faster???

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Rocks continued

• Now consider the two rocks held together by a
  length of string.
• If the smaller rock were to fall slower, it would
  impede the rate at which both rocks would fall.
• But the two rocks together would actually have
  more mass, and should therefore fall faster.
• A conundrum????? The previously held views could
  not have been correct.

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• Galileo had data which proved Aristotle and the
  church wrong, but church leaders were hardly
  moved in their position that all objects have
  their correct position in the world
• Furthermore, the use of Simplicio (or simpleton)
  as the head of the church in his dialog, was a
  direct insult to the church leaders themselves.