Motion graphs

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Motion graphs
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Why Graph?
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What is the best way to describe motion of an
object to somebody that did not witness it?

• With accuracy
• Can be understood by everybody in the world

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Describe my motion

• Write a description of my motion on a piece of
  paper

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A better way

• Represent the motion in the universal language
  Math
• Draw a graph

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Reasons

• Universal language
• Convey more information visually
• Less time to draw than write

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Graph Set up
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Axis

• What do the numbers mean

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Direction

• Is direction of motion indicated on the graph?

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Position-time grpahs
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Position-time graphs

• Position value is recorded as the vertical
  (y) component
• Time value is recorded as the horizontal
• The point (4,-8) means you are at -8 units from
  the central reference point at the 4 second mark

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Position-time graph

• Where is the object at the 3 second mark?
• 9 second mark?

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What information does the graph tell you about
motion

• Shape of the line
• Straight-vs-curved
• Tilt of the line
• Flat-vs-slanted
• Tilted upward-vs-tilted downward
• Vertical lines
• Placement of the line
• Start position
• Postive-vs negative territory

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Position Time graphs
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Position Time graphs

• Graph indicates
• Positive motion
• Constant velocity
• Between fast and slow rate of motion

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Moving very fast
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Moving very slow
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At rest

• How does a graph indicate that the object does
  not move

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Moving forward, backward at constant speed
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Which line shows no motion?Which line shows
fastest rate of motion?
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Speeding up
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• Graph indicates
• Non-constant velocity
• Positive motion
• Getting faster

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• Graph indicates
• Negative motion
• Non-constant velocity
• Getting faster

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Slowing down
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Slowing down moving forward
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• Graph indicates
• Negative motion
• Non-constant velocity
• Slowing down

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Forward, backward, or stopped

• If the graph is horizontal, then no motion has
  occurred.
• The position (vertical) value did not change over
  time
• If the final position is more positive than the
  initial position , it moved forward
• If the final position is less positive than
  initial position, it moved backward

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Is the object moving at a constant rate?

• Constant velocity means the rate of motion does
  not change over time
• Graphs show constant velocity by creating a
  straight line.
• Angle (tilt) of line does not matter

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How fast is it going?

• Constant velocity can occur in any direction
• Being stopped gives you a constant velocity of
  zero (0).
• The tilt of a straight line will indicate
  forward, or backward motion
• Slope of the line is a measure of the objects
  velocity
• The amount of tilt will indicate how fast th
  object goes

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What if it is not a straight line?

• Then the velocity is not constant
• If the velocity changes over time the car
  accelerates
• Any change in velocity indicates acceleration
• On a P-T graph, acceleration is indicated by a
  curved line

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It is getting faster or slower?

• Tangent lines
• Pick 2 point along section of graph
• Draw tangent lines
• If the slope of line increases, then object is
  getting faster
• Divide section up into equal 2 blocks of time.
• Compare the displacement in each
• If amount of displacement increases, it is
  getting faster

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Tangent line

• Line that touches a graph at only one point

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Mathematical method to determine slope of tangent
lines

• Determine the rate of change
• Derivatives in calculus

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Drawing and analyzing Position time graphs
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Draw the graph

• Draw a graph that would represent the following
  motion
• Positive Motion
• Non-Constant Velocity
• Slowing Down

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Information from graphs

• For each section with the same type of motion,
  you should be able to determine
• Is the object moving forward, backward, or
  stopped
• Is the motion constant or not?
• If constant, is the rate of motion fast or slow?
• If not constant, is the object getting faster or
  slower?

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Position-Time
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2
5
3
6
1
2
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Position time graph

• Each point on the graph indicates the position of
  the object at a certain time
• Shows both distance and displacement
• Y-axis indicates position
• X-axis indicates time

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Start position
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Most graphs are made from a combination of
different types of motions
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What you should be able to tell me about the graph

• Whether object is moving or not
• Which direction it moves
• Whether motion is constant or not
• Whether object speeds up or slows

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Given the graph, describe the motion
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Given description of motion, produce
corresponding graph
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Creating a graph from written information

• Draw a copy on a piece of paper

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In the next graph

• You will create a graph that represents the
  following motion
• Section 1- starts at the -2 meter position and
  moves with slow positive constant velocity
• Section 2- moves with fast negative constant
  velocity

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Information for next graph

• Section 3- moves with positive non-constant
  velocity and is getting faster
• Section 4- moves with a constant velocity of zero
• Section 5 moves with a negative non-constant
  velocity and is slowing down

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Example 1

• Starting from a position of (-3).
• Object speeds up, moving forward to the origin
• Object maintains constant velocity moving
  forward, reaches (4)
• Object slows down, moving forward, reaches (6)
• Object Stops for several seconds
• Object speeds up moving backwards

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Assumption

• If the question does not specify times, assume
  that the displacement of interest is over the
  entire graph

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Determine the velocity from a position time graph
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Position time graph with number values
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What is the velocity of the car during the first
1.5 second?

• Is it constant
• Is it relatively fast or slow?
• How do you find its actual value?

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To answer the initial problem

• (3 0)(m) / (1.5 0)(s) 3 m/s
• Find the rest of the constant velocity values
  shown on the graph

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Calculation of a constant velocity

• Slope of the line steepness
• To determine slope, find the rise over run
• Rise change in the y values between initial
  and final points Run change in the x values
• V (y2 y1) / (x2 x1)

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Slope of the line

• Constant velocity is demonstrated by a slanted
  straight line on a P-T graph
• The steepness indicates how fast the object moves
• To measure the steepness of a line, calculate the
  slope

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How to calculate the slope
Rise change in the position
Run change in the time
Slope Rise / Run
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Calculating the velocity

• Use (y2-y1) / (x2 x1) to calculate the slope
  (velocity)
• Organization of information
• Starts with identification of x and y values

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Calculate velocity on each section
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Determine instantaneous velocity from
Position-time graph
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Calculating the instantaneous velocity value

• For any section on a graph with constant
  velocity, all points in time within that section
  have the same velocity value.

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Determine Distance and displacement from position
time graph
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Overall Displacement

• Look at the difference between final position and
  start position
• Look at nothing else!!!
• What is the overall displacement of the above
  graph?

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Determine overall displacement

• On a p-t graph
• Displacement the difference between the start
  position and final position for the portion of
  the graph you are interested in
• D y2 y1

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Example
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Find the overall displacement for

1. The entire graph
2. the first 20 seconds
3. The last 20 seconds

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Overall Distance

• Measure the change in position between each hill
  and valley of the graph
• Then add up the changes
• There are no subtractions!!
• What is the total distance of the given graphs?

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Velocity-time grpahs
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Site of airplane crash
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Crash site of flight 93
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Typical plane crash v-t graph
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Velocity- time set up

• The vertical axis is measured in m/s
• Positive number represent forward motion
• Negative numbers represent backward motion
• The horizontal axis is measured in s

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Horizontal lines on a V-T graph
Velocity (m/s)
Time (s)
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Determine direction on V-T graph

• Any portion of graph above the zero line denotes
  forward motion
• Any graph on line, means object at rest

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Tilted lines on V-T graphs
V (m/s)
T (m/s)
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Tilted lines on V-T graphs
V (m/s)
T (m/s)
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What do tilted lines mean on a V-T graph?

• The object is changing its velocity
• Acceleration
• How it accelerates is determined by
• Location of graph (Above, Below x-axis)
• Type of tilt

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Acceleration is

• A change in the velocity of the object
• Tilted lines on a V-T graph indicate acceleration
• Acceleration can be calculated by finding the
  slope value of these lines

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What kind of motion is indicated on this graph?
V (m/s)
T (s)
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Constant Velocity or Acceleration?

• Horizontal lines represent CV
• Slanted lines represent ACC

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Faster or slower?

• Tilt towards zero line slower
• Tilt away from zero line faster

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What would curves mean on a V-T graph

• Curves would indicate non-constant acceleration
• Think roller coaster rides, when you get pushed
  against the harness or back into the seat
• The Jerk

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Identify motion for each part of the V-T graph
V (m/s)
T (m/s)
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Create a V-t graph that

• Section 1 Shows constant motion of (-3) m/s for
  5 seconds.
• Section 2 Shows constant acceleration (slowing
  down in the negative direction)
• Section 3 Shows no movement for 5 seconds
• Section 4 Shows constant acceleration (speeding
  up in the positive direction)

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Answer
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Vertical Lines?
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Can there be a vertical line in a graph of a real
objects motion?
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No

• That would mean that the object is at several
  places at an instant of time

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Differences between Position and Velocity time
graphs
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Differences between V-t graphs and P-T graphs

• The vertical s tell how fast, not location
• V-t does not indicate start position
• Direction is indicated by position on graph, not
  slope
• Acceleration is indicated differently on each
  type of graph

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Calculate overall Displacement
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• On V-T graph, what is the area equal to?

5 m/s
10 s
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How do you find the overall Displacement,
Distance from a V-t graph?

• How far does an object travel if it moves at 5
  m/s for 10 seconds?
• V d / t ? d t x V ? 5 x 10
• 50 m

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Finding overall displacement from V-T graph

• How far does an object travel if it goes
• 3 m/s for 10 seconds?

V
3
t
10
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To find displacement

• To find displacement of any section
• Find the area under the graph
• Any section that is above the x-axis is positive
  displacement
• Any section below the x-axis is negative
  displacement
• Combine displacements for overall velocity

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Calculate overall velocity
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Overall velocity

• Velocity overall displacement divided by
  overall time
• How can one determine overall time?

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Example
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What is the overall velocity for

1. The entire graph
2. The first 6 seconds
3. The last 6 seconds

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Calculate the overall velocity

• Overall Velocity total displacement / time

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Calculate acceleration from graph
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Can calculate the acceleration of an object from
a velocity time graph

• Same way as figuring out the velocity from a
  position time graph

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Calculating acceleration by graph

• Steepness on a v-t graph indicates how quickly an
  object accelerates
• For a v-t graph
• slope of the line acceleration

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To calculate the average acceleration of a moving
object

• Acceleration change in velocity /
  elapsed time
• Change in velocity final velocity value /
  initial velocity value Vf - Vi / t

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Calculate acceleration

• The acceleration slope of line on velocity-time
  graph

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How far and fast did the object go in the first 4
seconds?
3 m/s
4 s
V
t
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To find distance

• All areas (above or below the x-axis) are
  positive distance values for each section
• Add up all sections

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Do Now

• Describe the motion that produced the following
  V-T graph

V (m/s)
4
30
t (s)
5
10
20
13
24
-5
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To find overall velocity from a V-t graph

• Total displacement / total time
• overall velocity

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Go back to the do now problem

• Find overall displacement
• Find overall distance
• Find overall velocity

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Answers

• Overall displacement
• (-12.5) 0 6 28 8 (-15) 14.5 m
• Overall distance
• (12.5) 0 6 28 8 (15) 69.5 m
• Overall velocity 14.5 / 30 0.48 m/s

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Finding Overall Velocityof the motion on a graph

• Is NOT equal to the average of the velocities of
  each section
• Is equal to the overall displacement divided by
  the elapsed time

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Rearrange the equation for average acceleration

• Assume acceleration is constant
• vf vi at t is the elapsed time

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Determining displacement from a velocity time
graph

• The area between the graph and the x-axis
  indicates the displacement of the object for a
  given time

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Create one type of graph from other

• Converting P-T to V-T
• The constant value that the slanted lines means
  has changed
• Less curves in V-T graphs
• Start, final position will not be defined in V-T
  graph, but acceleration can be calculated

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Conversion of graphs

• Converting V-t to P-T
• Start position is not defined
• More curves in final P-T graph
• End points must be calculated

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Graphs created from position points

• Alternative graphs

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Ink drop method of graphing

• Use something to mark the position of an object
  at a certain time interval during its motion

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When graphing is not needed
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Acceleration-time graphs
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Acceleration time graphs

• Horizontal lines represent constant acceleration
• Slanted and curved lines represent acceleration
  that is not constant
• Slanted lines do mean that the acceleration
  changes at a constant rate ( Car crashing into
  brick wall)

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Stopping and staring on the Medusa