



Velocity
is the rate of change of of the position of an object. 





This
is kind of like how many miles something moved in an hour. 


Or
how many feet the thing moved in a minute. 





Most
people know about something about velocity 


from
the readings on a speedometer in a car.



The
speedometer reads speed NOT velocity. 





Velocity
is a little different because it includes a direction. 


So
while 60 miles per hour is a speed, 


60
miles per hour to the west is a velocity. 





In
math talk, this is the change in position over the change in time. 





Using
X to stand for position and t to stand for time, we have: 











In
Calculus, we get to talk about velocity that is happening RIGHT NOW 


This
is called INSTANTANEOUS VELOCITY, 


and
is written like this: 











If
we have an equation for the position of an object 


for
any time that we might be given, 


The
derivative of that equation 


is
the acceleration equation of the object! 





Example: 





Say
that the position of an object in feet from the start point 


at
some number of seconds after we start, 


is
given by the equation: 











Where
t is the seconds that have passed since the start, 


and
X is the distance in feet from the start point. 





When
two seconds have passed since the start, t = 2 


Putting
2 into the equation and solving for X, we get: 











The
direction of this velocity vector, 


is
whatever direction we want X to mean. 


For
example, if X is north then X is south. 





Now
let's find the acceleration of the object when t = 2. 


First,
find the derivative of the original equation. 


The
original equation was: 











So
using the power rule, the derivative equation is: 











So
when t = 2, the acceleration of the object is: 











The
sign on the acceleration is negative, 


so
the acceleration is also in the negative X direction. 





You
might think that acceleration is ALWAYS 


in
the same direction as velocity, but it's not. 





Acceleration
is the rate at which the velocity is changing. 


The
velocity in this problem, was measured in feet moved per second. 


The
acceleration is how that velocity is changing as each unit of time
passes. 





So
in this problem when t = 2, 


the
acceleration of the object is 64 feet per second PER SECOND. 





copyright 2005 Bruce Kirkpatrick 
