Transcript mathematical model of motion ppt

Mathematical Model of Motion
Chapter 5
Velocity Equations
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Average velocity: v = Dd/Dt
To find the distance traveled with constant
or average velocity : d = vt
To find an object’s location along the x
axis: x = x0 + vt
Constant Acceleration Equations
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Final velocity after acceleration:
 v = v0 + at
Final position after acceleration
 Dd = ½ (v + v0)t
2
 Dd = v0t + ½at
Final velocity after acceleration
 v2 = v02 + 2a(Dd)
Free Fall
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Objects accelerate as they fall
Acceleration due to gravity is
9.80 meters per second per
second
g = 9.80 m/s2
So speed of falling object
increases almost 10 m/s every
second it falls
Free Fall Calculations
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Instantaneous speed of
falling object = gt
acceleration times the
elapsed time of fall
To find distance of fall, d
= 1/2 gt2
In any time interval,
distance fallen = avg.
speed times elapsed time
Terminal Velocity
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Air resistance slows objects, depending on
surface area and air density
Terminal velocity (speed) is reached when
force of air resistance equals force due to
gravity
In vacuum, all objects fall at same rate
Graphing Motion: position vs. time
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Plot position (dependent variable) versus
time (independent variable)
Sometimes d is used to represent position
In horizontal motion, x is often used
instead
To find position at any time, read the
values off the graph using the line
Graphing Motion: Position Graph
with 2 or more objects
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Motion for two or more objects can be
plotted on the same graph
Where lines cross, objects are at the same
position
Graphing Motion: Position Graph
with Uniform Motion
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Equal displacements in equal time
intervals
Produces linear graph
Slope of position graph gives velocity
Horizontal line means zero velocity
Upward slope means positive velocity
Downward slope means negative velocity
Position Graph with Uniform
Motion
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Motion diagram to graph
Slope of a Graph
Graphing Motion: Position Graph
with Acceleration
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Graph will be curved
Constant acceleration gives parabola
Slope of graph at any point gives
instantaneous velocity
Position Graph with Acceleration
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Converting motion diagram to graph
Velocity vs Time Graphs
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Plot velocity as dependent variable, time
as independent variable
Constant velocity gives horizontal line
Slope of graph gives acceleration
Positive acceleration gives upward slope
Negative acceleration slopes downward
If graph is linear, acceleration is constant
Velocity vs Time: acceleration
Comparing Uniform and
Accelerated Motion Graphs
Uniform motion
Accelerated Motion
Comparing Positive and Negative
Velocity
Speeding up and Slowing Down
Velocity vs Time Graphs: Finding
Displacement
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Displacement can be found from velocity
graph by finding the area between the
graph and the time axis
Divide the area bounded by the graph line,
the horizontal axis and the initial and final
times into geometric sections (squares,
rectangles, triangles) and find the area
Area below the time axis is negative
displacement
Area under the Velocity Graph
Area Under the Velocity Graph
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Divide complex areas into triangles and
rectangles