Calculating with constant acceleration

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 $\hat{i}$ or $\hat{j}$ 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

$$a_0 = \frac{\Delta v}{\Delta t}$$

or

$$a_0 = \frac{v_2 - v_1}{\Delta t}$$

where $v_1$ is our velocity at the beginning of the time interval (initial velocity) and $v_2$ is our velocity at the end of the time interval (final velocity).

If we solve for the final velocity we get

$$v_2 = v_1 + a_0 (t_2 - t_1)$$

Since we have selected particular times, everything in this equation is a  constant. But we can be choose one or more of them to turn back into a variable.  In particular, let's call $t_2$, "$t$", and treat it like an independent variable.  Then we can see what the velocity looks like as a function of time. The "final velocity", since it corresponds to the velocity at time $t_2$ will become a dependent variable, $v$.

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 t₁ and some later time t that is unspecified)

$$v(t) = v_t + a_0 (t - t_1)$$

If we plot this velocity as a function of $t$, we get a figure that looks like the one at the left in the figures below.  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,  if a is constant, then

$$\langle v \rangle = \frac{v_1 + v_2}{2}$$

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:

$$\Delta v = a \Delta t$$

$$\Delta x = \langle v \rangle \Delta t$$

$$\Delta v = v_2 - v_1$$

$$\langle v \rangle =  \frac{v_1 + v_2}{2}$$

These 4 equations each have a clear and straightforward conceptual meaning and let you calculate whatever you need from whatever you know when the acceleration is constant. The tricky part sometimes, if remembering which equation works for $\Delta v$ and which for $\langle v \rangle$. It's easy to keep it straight if you keep the meaning of "change" and "average" in mind and think about what they would each be if the velocity were constant.

Workout: Calculating with constant acceleration

Joe Redish 9/10/11