Graphical Analysis of Motion

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Graphical Analysis of Motion

• Physics I
• Ms. Kniff

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Analyzing Motion Using Graphs

• Distance vs. Time Graphs
• Velocity vs. Time Graphs
• Acceleration vs. Time Graphs
• and..how the graphs are related
• and..what the graphs tell us

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Case 1 Truck Standing Still

• d
  vs. t

distance
time
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How do these graphs relate?
Remember v Dd/D t slope rise/run
for a d vs. t graph. slope Dd/Dt
velocity Therefore, the slope of a d vs. t graph
is velocity.
And.. a Dv/Dt slope rise/run
for a v vs. t graph. slope Dv/Dt
acceleration Therefore, the slope of a v vs. t
graph is acceleration.
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Case 1 Truck Standing Still

• v vs. t a
  vs. t

acceleration
velocity
time
time
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Case 2 Truck Moving Forward at a Constant
Velocity

• 
  d vs. t

distance
time
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How do these graphs relate?

• The slope of a d vs. t graph is velocity.
• The slope of a v vs. t graph is acceleration.

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Case 2 Truck Moving Forward at a Constant
Velocity

• v vs. t a vs. t

velocity
acceleration
time
time
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Case 3 Truck Accelerating Forward

• d vs. t

distance
time
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How do these graphs relate?

• The slope of a d vs. t graph is velocity.
• The slope of a v vs. t graph is acceleration.

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Case 3 Truck Accelerating Forward

• v vs. t a
  vs. t

acceleration
velocity
time
time
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EXAMPLE - Summarizing Cases 1,2,3
d vs t
B
C
distance
A
D
time

• AB - Constant Velocity Forward
• BC - No distance - Stopped
• CD - Neg Slope - Const Vel Backward

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EXAMPLE - Summarizing Cases 1,2,3(what if this
graph was v vs. t?)
v vs t
B
C
velocity
A
D
time

• AB - Inc.Velocity or Acceleration Forward
• BC - Constant Velocity
• CD - Dec.Velocity or Decel. Forward

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Is this everything that we need to know? What if
we change direction? (Sign Convention)
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Distance vs. TimeConstant Velocity (Sign
Convention)
Const. Vel. Forward
Stopped

Const. Vel. Backward
d
t
Const. Vel. Forward
-
Stopped
Const. Vel. Backward
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Distance vs. TimeAccelerated Motion (Sign
Convention)
Accelerating Forward ( )
Decelerating Forward (- -)

d
t
-
Decelerating Backward (- - )
Accelerating Backward ( - -)
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Velocity vs. Time (Sign Convention)
Inc. Vel. (Accel) Forward

forward
Const. Vel. Forward
Dec. Vel. (Decel) Forward
v
t
Dec. Vel. (Decel.) Backward
Const. Vel. Backward
backward
Inc. Vel. (Accel.) Backward
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EXAMPLE Complete Trip Graph
d
c
d
b
e
a
h

• a) constant velocity forward
• b) deceleration forward
• c) stopped
• d) constant velocity forward
• e) constant velocity backward
• f) deceleration backward
• g) constant velocity forward
• h) stopped at pt of origin

t
f
g
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Lets Review
d
Case 1 STOPPED
t
Slope
v
t
a
t
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Lets Review
d
Case 2 Constant Velocity
t
Slope
v
t
a
t
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Lets Review
Its more difficult to find the slope of a curve,
draw a tangent to the curve and find the slope of
the tangent line.
d
Case 3 Uniform Acceleration
t
Slope
v
t
a
t
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EXAMPLE
d
t
Which line represents the larger velocity?
The steeper the line, the larger the slope,
therefore the greater the velocity.
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Area of a Motion Time Graph
We can also use the area under the curve to
relate these graphs.
a
t
Area of a rectangle (l)(w) Area of an
acceleration vs time graph Area (a)(t)
(v/t)(t) v Therefore, the area under an a vs. t
graph is the velocity.
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Area of a Motion Time Graph
We can also use the area under the curve to
relate these graphs.
v
t
Area of a triangle ½(b)(h) Area of a velocity vs
time graph Area ½(t)(v) ½(t)(at) ½at2
d Therefore, the area under a v vs. t graph is
the distance.
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Lets Summarize Slope and Area
d
t
slope
area
v
t
a
t
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Analyzing Motion Graphs - REVIEWPosition vs.
Time Graphs

• Displacement is read from the vertical scale.
• Velocity is found from the slope of the graph.
• Average velocity is the slope of the line
  connecting two points on the graph.
• Instantaneous velocity is the slope of a line
  drawn tangent to the curve at a specific time.

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Analyzing Motion Graphs - REVIEWVelocity vs.
Time Graphs

• Displacement is the area under the graph.
• Velocity is read from the vertical scale.
• Acceleration is found from the slope of the
  graph.
• Average acceleration is the slope of the line
  connecting two points on the graph.
• Instantaneous acceleration is the slope of a line
  drawn tangent to the curve at a specific time.

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Analyzing Motion Graphs - REVIEWAcceleration vs.
Time Graphs

• Velocity is the area under the graph.
• Acceleration is read from the vertical scale.

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Now that we know how it works..

• Any questions?
• Lets try some examples.