Displacement, Velocity, and Acceleration and Their Graphical Representations

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• Displacement, Velocity, and Acceleration and
  Their Graphical Representations

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How can we describe motion?

• motion is the state of an object whose position
  is changing
• We quantify motion by describing how far an
  object moves in a particular amount of time
• Distance/Displacement measures how far something
  moves
• Speed/Veolcity measures how fast something is
  moving
• Acceleration measures the rate at which an
  objects velocity is changing

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Important Distinctions!

• Distance
• The total length of a path followed by a moving
  obeject
• Average Speed
• The ratio of the distance travelled to the time
  required to do so

• Displacement
• The net change in position of a moving object
• Average Velocity
• The ratio of the objects displacement to the
  elapsed time required for that displacement

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Graphical Description of Motion

• We can understand important aspects of an
  objects motion by studying graphs of its
  position, velocity, and acceleration as a
  function of time.
• Position versus time x vs. t
• Velocity versus time v vs t
• Acceleration versus time a vs t

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Constant position(not moving!)
Position (m)
In each case, the objects displacement is 0
since its final position is the same as its
initial position!
Time (s)
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Constant speed / velocity
v 1 m/s
Position (m)
Time (s)
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Constant velocity (v 1 m/s)
slope of (x,t) graph!
4
8
x (m)
4
2
8
t (s)
2
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Constant velocity (starting at x0)
Position vs time
Velocity vs time
v (m/s)
x (m)
The slope of a (v,t) graph is 0 if the velocity
is constant!
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v 1m/s starting at x 4 m
Slope is the same, but position is shifted by 4 m.
x (m)
t (s)
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Constant velocity (starting at x4 m)
Position vs time
Velocity vs time
v (m/s)
x (m)
Initial position does not change the slope
(velocity is the same)!
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Constant (uniform) acceleration
Uniform acceleration is defined as
a 1 m/s2
v (m)
Velocity changes by a fixed amount over equal
time intervals
Acceleration is the slope of a v,t graph!
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Position graph from uniform acceleration
v (m)
x (m)
Time (s)
Velocity is higher over each interval, so
distance traveled is greater!
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Velocity graph from position graph
8 s
slope 8
v (m)
6 s
slope 6
x (m)
Time (s)
Velocity AT 2 s is the slope of the tangent line
at 2 s
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Summary Graphical description of motion

• The slope of an (x,t) graph is the velocity
• Constant velocity constant slope
• Constant acceleration changing slope
• The slope of a (v,t) graph is the acceleration
• Constant acceleration constant slope
• If something is constant (position, velocity,
  acceleration), the slope of its graph is 0.