Class content I > The Main Question: Motion > Kinematics > Kinematic Variables > Acceleration
3.2.2.3
Prerequisites
If we have a constant acceleration  the velocity is either increasing or decreasing at a uniform rate  then we can figure out a lot of useful stuff. For this section, we'll assume that we are moving along a line. We could be traveling back and forth, but we won't consider changes of the angle of our motion. So we will drop the ihat or jhat that picks out which line we are traveling along and work only with one coordinate. For convenience here, we will call it x.
The velocity
If the acceleration is constant, then the velocity is changing at a constant rate. Our equation defining the acceleration is then
a_{0} = Δv/Δt
or
a_{0} = (v_{2}  v_{1})/Δt
where v_{1} is our velocity at the beginning of the time interval and v_{2} is our velocity at the end of the time interval.
If we solve for the final velocity we get
v_{2} = v_{1} + a_{0} (t_{2}  t_{1})
These constants can be chosen to move. In particular, let's call t_{2}, "t", and treat it like a variable. Then we can see what the velocity looks like as a function of time.
If this seems a little weird to you, get used to it. In physics, since we tend to describe both general and specific situations, we will often let a constant "wander around", becoming a variable. And we will often fix a variable at a specific value in order to have an equation apply to a specific situation.


This gives us the equation for v (which holds between time t_{1} and some later time t that is unspecified)
v(t) = v_{1} + a_{0} (t  t_{1})
If we plot this, we get a figure that looks like the one at the left. If we want to figure out what the average velocity in that time interval is, we have to find the constant line so that the areas under the true velocity curve equals the area under the average velocity line (a constant). We do this by adjusting the average velocity line so that the light pink area (the area no longer included) and the light blue area (the extra area now included) are equal.
It's pretty clear both from thinking about how the velocity changes and from looking at the graph, that the average velocity is going to be halfway between the endpoints; that is,
<v> = (v_{1} + v_{2})/2 (if a is constant).
Since we know that the definition of the average velocity is "that velocity which, if you go constantly with it, will produce the same displacement", we can see that if the acceleration is constant:
Δv = a Δt or v_{2}  v_{1} = a Δt
<v>=(v_{1} + v_{2})/2
Δx = <v>Δt
These three equations (or 4 if you count knowing what a delta of something means) each have a clear and straightforward conceptual meaning and let you calculate whatever you need from whatever you know when the acceleration is constant.
Joe Redish 9/10/11
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