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Get Your Motor Runnin... Head Out On The Highway
Velocity & Acceleration Vectors

 
 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

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