Motion In One Dimension

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Motion In One Dimension
PLATO AND ARISTOTLE
GALILEO GALILEI
LEANING TOWER OF PISA
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Distance, Position and Displacement
Distance (d)
Distance is the total length of a path traveled
by an object.
Distance is always positive, even if an object
reverses its direction.
Position (x or y)
Position is the location of an object relative to
an origin.
Position can be positive or negative.
1. What is the distance traveled if an object
starts at point C, moves to A, then to B?
12 units
Displacement (?x or ?y)
Displacement is the change in position of an
object.
2. What are the positions of objects located at
points A, B, and C relative to the origin?
-5, -2, 4 units
3. What is the displacement of an object that
starts at point C and moves to point B?
-2 (4) -6 units
4. What is the displacement of an object that
starts at point A, then moves to point C and then
moves to point B?
Displacement can be positive or negative.
-2 (-5) 3 units
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Distance and Position Graphs
Distance vs. Time
Position vs. Time
d (m)
x (m)
positive
constant positive velocity
t (s)
constant speed
negative
constant negative velocity
t (s)
Distance graphs show how far an object travels.
Speed is determined from the slope of the graph,
which can only be positive.
Position graphs show how far, and in which
direction, an object travels. Velocity (speed
with direction) is determined from the slope of
the graph.
Notice that these graphs show constant speed.
(How do you know?)
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Average Speed vs. Average Velocity
Average speed is the distance traveled divided by
time elapsed.
Average velocity is displacement divided by time
elapsed.
Example A sprinter runs 100 meters in 10
seconds, and then walks slowly back to the
starting blocks in 30 seconds. What is the
sprinters average speed and average velocity for
the entire time?
x (m)
d (m)
slope speed
slope velocity
t (s)
t (s)
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Instantaneous Speed and Velocity
Instantaneous speed is the speed of an object at
an exact moment in time. Instantaneous velocity
includes direction too.
Instantaneous speed (or velocity) is found
graphically from the slope of a tangent line at
any point on a distance (or position) vs. time
graph.
negative tangent slope negative velocity
x (m)
d (m)
slope of tangent speed
t (s)
t (s)
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Velocity and Displacement (Honors)
Velocity vs. Time
v (m/s)
t (s)
area displacement (.5)(3 s)(30 m/s) (4
s)(30 m/s) (.5)(1 s)(30 m/s) 180 m
For a non-linear velocity graph, the area can be
determined by adding up infinitely many pieces
each of infinitely small area, resulting in a
finite total area! This process is now known as
integration, and the function is called an
integral.
A velocity graph can be used to determine the
displacement (change in position) of an
object. The area of the velocity graph equals the
objects displacement.
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The Physics of Acceleration
Acceleration is how quickly how fast changes
PAUL HEWITT, CITY COLLEGE, S.F.
how fast
means velocity
how fast changes
means change in velocity
how quickly
mean how much time elapses
Acceleration is defined as the rate at which an
objects velocity changes.
Metric (SI) units
Acceleration has units of meters per second per
second, or m/s/s, or m/s2.
Acceleration is considered as a rate of a rate.
Why?
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Types of Acceleration
Velocity vs. Time
Velocity vs. Time
v (m/s)
v (m/s)
slope average acceleration
slope acceleration
slope instantaneous acceleration
t (s)
t (s)
Constant Acceleration
Varying Acceleration
Average acceleration is the slope of a secant
line for a velocity vs. time graph. Instantaneous
acceleration is the slope of a tangent line for a
velocity vs. time graph. (Compare to, but DO NOT
confuse with average and instantaneous velocity
on a position vs. time graph.)
Constant acceleration is the slope of the line
for a velocity vs. time graph. (Compare to, but
DO NOT confuse with constant velocity on a
position vs. time graph.)
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An Acceleration Analogy
Compare the graph of wage versus time to a
velocity versus time graph. The slope of the wage
graph is wage change rate. Slope of the
velocity graph is acceleration. What is the slope
for each graph, including units? In this case the
wage change rate is constant. The graph is
linear because the rate at which the wage changes
is itself unchanging (constant)! The analogy
helps distinguish velocity from acceleration
because it is clear that wage and wage change
rate (acceleration) are different.
slope acceleration 1 m/s/s
slope wage change rate
1//hr/month
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An Acceleration Analogy
Earnings, Wage, and Wage Change Rate
Position, Velocity, and Acceleration
Can a person have a high wage, but a low wage
change rate?
Can an object have a high velocity, but a low
acceleration?
Making good hourly money, but getting very small
raises over time.
Moving fast, but only getting a little faster
over time.
Can a person have a low wage, but a high wage
change rate?
Can an object have a low velocity, but a high
acceleration?
Making little per hour, but getting very large
raises quickly over time.
Moving slowly, but getting a lot faster quickly
over time.
Can a person have a positive wage, but a negative
wage change rate?
Can an object have a positive velocity, but a
negative acceleration?
Making money, but getting cuts in wage over time.
Moving forward, but slowing down over time.
Can a person have zero wage, but still have wage
change rate?
Can an object have zero velocity, but still have
acceleration?
Making no money (internship?), but eventually
working for money.
At rest for an instant, but then immediately
beginning to move.
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Direction of Velocity and Acceleration
vi a motion
0
0
0
0




constant positive vel.
constant negative vel.
speeding up from rest
speeding up from rest
speeding up
speeding up
slowing down
slowing down
click for applet
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Kinematic Equations of Motion
Assuming constant acceleration, several equations
can be derived and used to solve motion problems
algebraically.
Slope equals acceleration
Velocity vs. Time (Constant Acceleration)
v (m/s)
Area equals displacement
vf
vi
t
Eliminate final velocity
t (s)
Eliminate time
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Freefall Acceleration
Aristole wrongly assumed that an object falls at
a rate proportional to its weight.
Galileo proved that all objects freefall (in a
vacuum, no air resistance) at the same rate.
An inclined plane reduced the effect of gravity,
showing that the displacement of an object is
proportional to the square of time.
click for video
Location g
Equator 9.780
Honolulu 9.789
Denver 9.796
San Francisco 9.800
Munich 9.807
Leningrad 9.819
North Pole 9.832
Kinematic equations of freefall acceleration
Since the acceleration is constant, velocity is
proportional to time.
Latitude, altitude, geology affect g.
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A Velocity Analogy
Compare constant velocity (uniform motion) to
making money doing a job. Say you baby sit for
10/hour. Its easy to graph earnings as a
function of time. Compare the graph of earnings
versus time to position versus time. The slope of
the earnings graph is wage. Slope of the position
graph is velocity.
slope velocity 10 m/s
slope wage 10//hr
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A Velocity Analogy
Compare constant acceleration to getting raises
while doing a job. Maybe you baby sit for
10/hour, but now you get regular wages
increases. Compare the graph of earnings versus
time to position versus time. Slope of a secant
line is the average wage (compare with average
velocity.) Slope of a tangent line is the
instantaneous wage (compare with instant.
velocity.)