Kinematics in one dimension

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Chapter 2

• Kinematics in one dimension

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Kinematics

• Description of how objects move
• One dimensional translational motion is an object
  moving along a straight line path
• We will consider objects to be point particles
  for simplicity (think about a runner

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In Other Words

• A complex look at something moving in a straight
  line.
• This complexity is necessary for later use in the
  study of velocity and acceleration
• You will eventually be able to solve the
  velocity, time, or position in the x and y
  directions at any point along this path.

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Frame of Reference
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Reference Frames and Displacement
We make a distinction between distance and
displacement. Displacement (blue line) is how
far the object is from its starting point,
regardless of how it got there. Distance traveled
(dashed line) is measured along the actual path.
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Distance vs. Displacement

• Mary Prakel starts at the finish line, makes 8
  grueling laps around the track, and stops exactly
  where she started.
• What is her distance traveled?
• What is her displacement?

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Coordinate axes

• Used to show direction
• Position of an object at any moment is given by
  its x coordinate

• Directions of the axes can be placed at our
  convenience.
• Vertical motion often represented by the y axis

y
x
-x
-y
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Vectors

• Quantities that have both magnitude and
  direction.
• Ex displacement, velocity, acceleration
• Quantities such as distance and speed, which
  describe only magnitude, are called scalars.

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Vectors continued

• Vectors which point in one direction get a
  positive sign
• Those in the opposite direction get a negative
  sign.
• Just because a displacement vector is negative
  does not mean it is found in the x side of the
  line. (what would be an example of this?)

y
x
x2
x1
x
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Vectors continued

• Assume each division stands for 5 m
• Displacement is solved using the formula ?x x2
  x1
• ?x 20m 5m
• Displacement is 15 m

y
x
x2
x1
x
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Vectors continued

• This vector shows a displacement of the same
  magnitude in the opposite direction.
• ?x x2 x1 in this case is
• ?x 5m 20m -15m
• Displacement is -15m

y
x2
x1
x
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Vectors continued

• In these cases a vector pointing right is
  positive while a vector pointing left is
  negative.

y
x
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Understanding Quantities
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Speed vs. Velocity

• Speed refers to how far an object travels in a
  certain amount of time regardless of direction.
• Average speed distance traveled
• time elapsed
• Speed is always a positive number

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Average Velocity

• Velocity refers to an objects displacement in a
  certain amount of time.
• Rate of change of position
• Average velocity displacement
• time elapsed
• final position initial position
• time elapsed
• Can be positive or negative

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Cruise control automatically keeps your car at a
constant speed. You head east for 4000 s with
the cruise control set at 31 m/s. You end up
150,000 m east of Versailles.

• How far east of Versailles did you start out?
• Sketch a graph of x vs. t for this motion.
• How fast are you traveling in mph?

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about 70 mi/hr
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Some Av. Velocity Examples
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Solve the Average Velocity for A,B,C,D
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Now draw x vs. t graphs for the four cars.
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Check your understanding

• Which of the cars have positive displacements?
• Which graph has the steepest slope?
• Put the cars speeds in order from greatest to
  least.
• Which has a greater velocity B or D?

• A and C (B and D have negative displacements)
• B (C has the most gradual slope)
• B, D, A, C.
• D has the greater velocity. A more negative
  number is not greater than a less negative number.

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Nonuniform Motion

• Description of an object that does not maintain
  the same motion during the entire time interval
  being studied
• In other words, it has changing velocity during
  the time being studied
• The previous graphs represented uniform motion.
  Lets look at some nonuniform motion.

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Nonuniform motion

• Attempt to solve the average velocity for each of
  these vehicles between 0 and 4 s.
• Then attempt to sketch a graph for each cars
  nonuniform motion. (not all graphs will be
  straight lines)

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Average velocity

• A is 50 m/s
• B is 50 m/s
• C is 50 m/s
• D is -50 m/s
• E is -50 m/s

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Interpreting slope for nonuniform motion

• If you draw a straight line segment (secant)
  connecting points P1 and P2, its slope is the
  average velocity of the object between those two
  points
• Which segment has the greater average velocity?

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Average vs. Instantaneous Velocity

• The average velocities we have been solving do
  not describe variations in the objects velocity
  throughout the trip.
• The graphs just made do not necessarily represent
  the actual velocities of the cars at any instant
  of time between 0 and 4 s.
• To find the velocity of an object at any moment
  during the trip, a speedometer, graphical
  analysis, or calculus must be used.

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2-3 Instantaneous Velocity
The instantaneous velocity is the average
velocity, in the limit as the time interval
becomes infinitesimally short.
(2-3)
These graphs show (a) constant velocity and (b)
varying velocity.
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Graphical analysis

• To graphically solve the instantaneous velocity
  at any point on the curve, we must draw a tangent
  line that just grazes the curve and then solve
  its slope.
• All curves are essentially made up of many
  straight lines.

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Graphs of v vs. t

• Graph a shows non uniform motion x vs. t.
• The lines of tangent for P1-P3 show the
  instantaneous velocity at those points.
• b shows the v vs. t graph for the same motion.

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Notice

• P1 slope is 0, so velocity is zero
• P2 has the steepest slope, so it represents the
  greatest velocity.
• P3 has a slope less than P2 so the velocity is
  less at P3 than at P2
• This is the graphical representation of a
  derivative in calculus.