Unit 01: 1D Kinematics

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Unit 011-D Kinematics

• How can an objects motion be described in words?
• How can various aspects of an objects motion be
  expressed in and determined from graphs?
• How can mathematical equations be used to model,
  calculate and predict aspects of an objects
  motion?

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Describing Motion
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Kinematics

• the method of describing the motion of objects
  using words, diagrams, numbers, graphs, and
  equations
• disregard what caused the motion
• GOAL develop graphical and mathematical models
  to describe and analyze motion

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Basic Kinematics Terminology

• Position objects location at any point in time
• relative to the origin (visualize an imaginary
  number line)
• symbol x (horizontal), y (vertical)
• units meters (m)
• Time duration of an event
• how long after the stopwatch was started
• symbol ?t
• units seconds (s)

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Basic Kinematics Terminology

• Displacement change in position
• has direction can be positive, negative, or zero
• symbol ?x
• ?x xf xi
• units meters (m)
• Distance how far an object travels
• no direction total path length
• symbol d
• units meters (m)

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Basic Kinematics Terminology

• Velocity rate of change in position
• depends on displacement and time
• has direction positive (forward), negative
  (backward)
• symbol v
• units meters per second (m/s)
• Speed rate at which distance is covered
• depends on distance and time
• no direction just how fast
• same symbol and units as velocity

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Describing Velocity (or speed)

• At Rest
• not moving remaining in the same position
• Constant Velocity
• object maintains the same velocity (ex. cruise
  control)
• equal increases (or decreases) in position in
  equal intervals of time

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Describing Velocity (or speed)

• Instantaneous Velocity
• velocity at any given moment
• for most objects, this is constantly changing
• Average Velocity
• average of all instantaneous velocities
• calculated from measurements of displacement and
  time

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Average Velocity (or speed)
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Average Velocity (or speed)
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Scalar vs. Vector Quantities

• Scalar Quantity
• fully described by magnitude alone
• no direction just a number
• Magnitude how much numerical value
• Examples so far
• Distance
• Speed
• Time

• Vector Quantity
• fully described by magnitude and direction
• a number plus direction
• Direction can be compass direction or / -
• Examples so far
• Displacement
• Velocity

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Example 1

• I walk 4 m East, 2 m South, 4 m West 2 m North.

What distance did I travel? What displacement did
I have?
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Example 2

• The diagram below shows the position of a
  cross-country skier at various times. At each of
  the indicated times, the skier turns around and
  reverses the direction of travel. In other words,
  the skier moves from A to B to C to D.
• What is the distance traveled by the skier? The
  displacement?
• What is the skiers average speed? Average
  velocity?

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Example 3

• Seymour Butz views football games from under the
  bleachers, pacing back and forth to get the best
  view. The diagram below shows several of
  Seymour's positions at various times. In other
  words, Seymour moves from position A to B to C to
  D.
• What is the distance traveled by the Seymour? The
  displacement?
• What is Seymours average speed? Average
  velocity?

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Graphs of Motion
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Position-Time Graphs

• Graph of position time data
• Graphed as (x,y) coordinate pairs
• Independent Variable (x-axis) Time
• Dependent Variable (y-axis) Position

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Position-Time Graphs
What do these graphs tell you about the objects
motion?
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Position-Time Graphs

• Working backwards, what is the meaning of the
  slope of a position-time graph?

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Position-Time Graphs
Determine the velocity from each of the above
graphs.
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Position-Time Graphs
What is happening to the velocity of an object
whose position-time graph looks like this? What
is this object doing?
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Acceleration

• Acceleration rate of change in velocity
• tells us how quickly something is changing
  velocity
• change in velocity can either be in magnitude or
  direction
• symbol a
• units m/s2
• Acceleration is a vector quantity. It has a
  direction!
• Positive acceleration velocity is increasing
• Negative acceleration velocity is decreasing
• Commonly called deceleration

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Acceleration
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Velocity-Time Graphs

• Graph of velocity time data
• Graphed as (x,y) coordinate pairs
• Independent Variable (x-axis) Time
• Dependent Variable (y-axis) Velocity

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Velocity-Time Graphs
What do these graphs tell you about the objects
velocity?
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Velocity-Time Graphs

• Working backwards, what is the meaning of the
  slope of a velocity-time graph?

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Velocity-Time Graphs
Determine the acceleration from each of the above
graphs.
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Velocity-Time Graphs
What is happening to the acceleration of an
object whose velocity-time graph looks like this?
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Velocity-Time Graphs

• An objects displacement be determined from a
  velocity-time graph by finding the area under the
  curve.
• Simplify to basic geometric shapes.
• Areas under the x-axis are negative and imply a
  negative displacement.
• Add displacements of all geometric shapes.

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Velocity-Time Graphs
Determine displacement from these velocity-time
graphs.
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Velocity-Time Graphs
How would you determine the displacement from
this graph?
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Graphs of Motion - Summary

• Position-Time Graphs
• slope velocity
• horizontal line at rest
• straight line w/ positive slope constant
  positive velocity
• straight line w/ negative slope constant
  negative velocity
• curved line acceleration
• Velocity-Time Graphs
• slope acceleration
• horizontal line constant velocity
• horizontal line on x-axis at rest
• straight line w/ positive slope constant
  positive acceleration
• straight line w/ negative slope constant
  negative acceleration
• curved line non-uniform acceleration
• area under curve displacement

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Mathematically ModelingUniformly-Accelerated
Motion
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Mathematical Models of Motion

• Constant acceleration is called uniform
  acceleration.
• Galileo defined uniform acceleration as equal
  increases in velocity in equal intervals of
  time.We will experimentally test this
  definition.
• Algebraic equations can be used to model the
  motion of a uniformly-accelerating object.
• Derived from basic acceleration average
  velocity equations.
• Show the relationships between acceleration,
  displacement, time, and change in velocity.

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Mathematical Models of Motion
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Acceleration Displacement

• Mathematical models can provide insight into the
  displacement of a uniformly-accelerating object
• Consider an object that accelerates from rest at
  10 m/s2.
• What is its displacement after 1 s?
• How much distance was covered in the first second
  alone?
• What is its displacement after 2 s?
• How much distance was covered in the second
  second alone?
• What is its displacement after 3 s?
• How much distance was covered in the third second
  alone?
• What is its displacement after 4 s?
• How much distance was covered in the fourth
  second alone?
• Do you see a pattern developing?

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Acceleration Displacement

• We can think of acceleration as covering
  increasing distances in equal intervals of time
• Or, putting it another way, acceleration is
  covering equal distances in decreasing intervals
  of time

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Example 1

• Ima Hurryin is approaching a stoplight moving
  with a velocity of 30.0 m/s. The light turns
  yellow, and Ima applies the brakes and skids to a
  stop.
• If Ima's acceleration is -8.00 m/s2, determine
  the displacement of the car during the skidding
  process.
• How long does it take Ima to stop?

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Example 2

• Ben Rushin is waiting at a stoplight. When it
  finally turns green, Ben accelerated from rest at
  a rate of a 6.00 m/s2 for 4.10 seconds.
• Determine the displacement of Ben's car during
  this time period.
• How fast is Ben going after this time?

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Free Fall

• Objects also accelerate uniformly when they rise
  or fall vertically through the air under the
  influence of gravity.
• Free Fall refers to the vertical motion (up
  and/or down) of an object under the influence of
  gravity ONLY. Air resistance is ignored.

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Free Fall

• All objects in Free Fall experience the same
  uniform acceleration due to gravity, regardless
  of shape, size, or mass.
• An object in free fall experiences a uniform
  acceleration of -9.8 m/s2. The negative means
  that this is a downward acceleration.
• By convention, up is the positive direction and
  down is negative.

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Free Fall

• The acceleration is always -9.8 m/s2, regardless
  of location.
• If an object is dropped the initial velocity of
  the object is 0 m/s.If it is thrown upwards, it
  has a positive initial velocity.
• At the peak of its motion, the velocity of the
  object is 0 m/s
• At the same height, an object will have the same
  velocity, regardless of whether it is going up or
  coming down.
• The same equations apply, except
• the acceleration is always -9.8 m/s2. Easy, huh?
• ?y is used instead of ?x

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Example 1

• Luke Autbeloe drops a pile of roof shingles from
  the top of a roof located 8.52 meters above the
  ground.
• Determine the time required for the shingles to
  reach the ground.
• How fast will the shingles be moving at the
  instant they hit the ground?

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Example 2

• Rex Things throws his mother's crystal vase
  upwards with an initial velocity of 26.2 m/s.
• Determine the height to which the vase will rise
  above its initial height.
• Assuming Rex decided to catch the vase, how long
  will it take to return to his hand?
• How fast will the vase be moving at the instant
  Rex catches it?