1-Dimensional Kinematics

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1-Dimensional Kinematics
Kinematics
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Average vs. Instantaneous Speed

• The speedometer of a car reveals information
  about the instantaneous speed of your car that
  is, it shows your speed at a particular instant
  in time.
• Average speed is a measure of the distance
  traveled in a given period of time it is
  sometimes refered to as the distance per time
  ratio

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The Stoplight

• A blue car moving at a constant speed of 10 m/s
  passes a red car that is at rest. This occurs at
  a stoplight the moment that the light turns
  green. The clock is reset to 0 seconds and the
  velocity-time data for both cars are collected
  and plotted. The red car accelerates from rest at
  4 m/s/s for three seconds and then maintains a
  constant speed. The blue car maintains a constant
  speed of 10 m/s for the entire 12 seconds.
  Observe the motion and make meaning of the
  accompanying graphs to answer the following
  questions
• What is the final velocity of a car that
  accelerates from rest at 4 m/s/s for three
  seconds?
• What is the displacement of each individual car
  after three seconds (consider a kinematic
  equation or the area of the velocity-time graph)?
  
• What is the slope of the line for the red car for
  the first three seconds?
• What is the displacement of each individual car
  after nine seconds (use the area of the
  velocity-time graph)?
• Does the red car pass the blue car at three
  seconds? If not, then when does the red car pass
  the blue car?
• When lines on a velocity-time graph intersect,
  does this mean that the two cars are passing by
  each other? If not, what does it mean?

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Positive Velocity and Positive Acceleration

• An object which moves in the positive direction
  has a positive velocity. If the object is
  speeding up, then its acceleration vector is
  directed in the same direction as its motion (in
  this case, a positive acceleration).
• The "ticker tape" shows that each consecutive dot
  is not the same distance apart (i.e., a changing
  velocity)
• The position-time graph shows that the slope is
  changing (meaning a changing velocity) and
  positive (meaning a positive velocity). The
  velocity-time graph shows a line with a positive
  (upward) slope (meaning that there is a positive
  acceleration) the line is located in the
  positive region of the graph (corresponding to a
  positive velocity). The acceleration-time graph
  shows a horizontal line in the positive region of
  the graph (meaning a positive acceleration).

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The Passing Lane

• Observe the two cars below. The blue car starts
  "ahead of" the red car (which actually starts
  "off the screen"). Since the red car is moving
  faster, it eventually catches up with and passes
  the blue car. Observe the velocity-time graphs
  for these two cars. Each car's motion is
  represented by a horizontal line, indicating a
  constant velocity. Observe that even though the
  cars pass each other, the lines on the
  velocity-time graphs do not intersect. Since the
  cars never have the same velocity, the lines on
  the velocity-time graph never cross. The lines
  would intersect for a position vs. time graph
  the fact that the red car passes the blue car
  means that there is an instant in which they
  occupy the same position. The two cars have the
  same position at seven seconds yet they never
  have the same velocity at any instant in time.

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Negative Velocity and Negative Acceleration

• Observe that the object below moves in the
  negative direction with a changing velocity. An
  object which moves in the negative direction has
  a negative velocity. If the object is speeding up
  then its acceleration vector is directed in the
  same direction as its motion (in this case, a
  negative acceleration). The "ticker tape" shows
  that each consecutive dot is not the same
  distance apart (i.e., a changing velocity). The
  position-time graph shows that the slope is
  changing (meaning a changing velocity) and
  negative (meaning a negative velocity). The
  velocity-time graph shows a line with a negative
  (downward) slope (meaning that there is a
  negative acceleration) the line is located in
  the negative region of the graph (corresponding
  to a negative velocity). The acceleration-time
  graph shows a horizontal line in the negative
  region of the graph (meaning a negative
  acceleration).

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Constant Positve Velocity

• Observe that the object below moves with a
  constant velocity in the positive direction. The
  "ticker tape" shows that each consecutive dot is
  the same distance apart (i.e., a constant
  velocity). The position-time graph shows that the
  slope is both constant (meaning a constant
  velocity) and positive (meaning a positive
  velocity). The velocity-time graph shows a
  horizontal line with zero slope (meaning that
  there is zero acceleration) the line is located
  in the positive region of the graph
  (corresponding to a positive velocity). The
  acceleration-time graph shows a horizontal line
  at the zero mark (meaning zero acceleration).

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Direction of Acceleration and Velocity

• Consider the motion of a Hot Wheels car down an
  incline, across a level, straight section of
  track, around a 180-degree curve, and finally
  along a final straight section of track. Such a
  motion is depicted in the animation below. The
  car gains speed while moving down the incline -
  that is, it accelerates. Along the straight
  sections of track, the car slows down slightly
  (due to air resistance forces) again the car
  could be described as having an acceleration (or
  perhaps you prefer deceleration). Finally, along
  the 180-degree curve, the car is changing its
  direction once more the car is said to have an
  acceleration due to the change in the direction.
  Accelerating objects have a changing velocity -
  either due to a speed change (speeding up or
  slowing down) or a direction change.

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Acceleration

• Observe the animation of the three cars below.
  Which car or cars (red, green, and/or blue) are
  undergoing an acceleration? Study each car
  individually in order to determine the answer. If
  necessary, review the definition of acceleration.

As a final test of your understanding, consider
the position-time graph at the right. Each one of
the three lines on the position-time graph
corresponds to the motion of one of the three
cars. Match the appropriate line to the
particular color of car.
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Negative Velocity and Positive Acceleration

• Observe that the object below moves in the
  negative direction with a changing velocity. An
  object which moves in the negative direction has
  a negative velocity. If the object is slowing
  down then its acceleration vector is directed in
  the opposite direction as its motion (in this
  case, a positive acceleration). The "ticker tape"
  shows that each consecutive dot is not the same
  distance apart (i.e., a changing velocity). The
  position-time graph shows that the slope is
  changing (meaning a changing velocity) and
  negative (meaning a negative velocity). The
  velocity-time graph shows a line with a positive
  (upward) slope (meaning that there is a positive
  acceleration) the line is located in the
  negative region of the graph (corresponding to a
  negative velocity). The acceleration-time graph
  shows a horizontal line in the positive region of
  the graph (meaning a positive acceleration).

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Positive Velocity and Negative Acceleration

• Observe that the object below moves in the
  positive direction with a changing velocity. An
  object which moves in the positive direction has
  a positive velocity. If the object is slowing
  down then its acceleration vector is directed in
  the opposite direction as its motion (in this
  case, a negative acceleration). The "ticker tape"
  shows that each consecutive dot is not the same
  distance apart (i.e., a changing velocity). The
  position-time graph shows that the slope is
  changing (meaning a changing velocity) and
  positive (meaning a positive velocity). The
  velocity-time graph shows a line with a negative
  (downward) slope (meaning that there is a
  negative acceleration) the line is located in
  the positive region of the graph (corresponding
  to a positive velocity). The acceleration-time
  graph shows a horizontal line in the negative
  region of the graph (meaning a negative
  acceleration).

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Constant Negative Velocity

• Observe that the object below moves with a
  constant velocity in the negative direction. The
  "ticker tape" shows that each consecutive dot is
  the same distance apart (i.e., a constant
  velocity). The position-time graph shows that the
  slope is both constant (meaning a constant
  velocity) and negative (meaning a negative
  velocity). The velocity-time graph shows a
  horizontal line with zero slope (meaning that
  there is zero acceleration) the line is located
  in the negative region of the graph
  (corresponding to a negative velocity). The
  acceleration-time graph shows a horizontal line
  at the zero mark (meaning zero acceleration).

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Two-Stage Rocket

• Observe the motion of the two-stage rocket and
  the corresponding velocity-time graph below. The
  rocket has two consecutive fuel stages followed
  by a free-fall motion (no fuel). In the two fuel
  stages, the rocket experiences an upward
  acceleration of 10 m/s/s and 4.29 m/s/s
  respectively. This acceleration is depicted by
  the slope on the velocity-time graph. After ten
  seconds, the second fuel stage ends and the
  rocket is acted upon only by the force of
  gravity. It subsequently experiences a downward
  acceleration of -10 m/s/s. Note however, that
  from 10 to 16 seconds, the rocket continues
  moving upward (the velocity values are positive).
  During these six seconds, the rocket is moving
  upward but slowing down (the acceleration is
  downwards or negative as denoted by the
  negatively-sloped line). It is not until after
  t16 seconds that the rocket begins to move
  downwards.

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