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Velocity
is the rate of change of of the position of an object. |
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This
is kind of like how many miles something moved in an hour. |
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Or
how many feet the thing moved in a minute. |
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Most
people know about something about velocity |
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from
the readings on a speedometer in a car.
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The
speedometer reads speed NOT velocity. |
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Velocity
is a little different because it includes a direction. |
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So
while 60 miles per hour is a speed, |
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60
miles per hour to the west is a velocity. |
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In
math talk, this is the change in position over the change in time. |
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Using
X to stand for position and t to stand for time, we have: |
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In
Calculus, we get to talk about velocity that is happening RIGHT NOW |
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This
is called INSTANTANEOUS VELOCITY, |
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and
is written like this: |
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If
we have an equation for the position of an object |
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for
any time that we might be given, |
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The
derivative of that equation |
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is
the acceleration equation of the object! |
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Example: |
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Say
that the position of an object in feet from the start point |
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at
some number of seconds after we start, |
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is
given by the equation: |
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Where
t is the seconds that have passed since the start, |
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and
X is the distance in feet from the start point. |
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When
two seconds have passed since the start, t = 2 |
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Putting
2 into the equation and solving for X, we get: |
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The
direction of this velocity vector, |
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is
whatever direction we want -X to mean. |
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For
example, if X is north then -X is south. |
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Now
let's find the acceleration of the object when t = 2. |
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First,
find the derivative of the original equation. |
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The
original equation was: |
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So
using the power rule, the derivative equation is: |
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So
when t = 2, the acceleration of the object is: |
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The
sign on the acceleration is negative, |
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so
the acceleration is also in the negative X direction. |
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You
might think that acceleration is ALWAYS |
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in
the same direction as velocity, but it's not. |
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Acceleration
is the rate at which the velocity is changing. |
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The
velocity in this problem, was measured in feet moved per second. |
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The
acceleration is how that velocity is changing as each unit of time
passes. |
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So
in this problem when t = 2, |
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the
acceleration of the object is -64 feet per second PER SECOND. |
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copyright 2005 Bruce Kirkpatrick |
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