Deriving the constant-acceleration equations
When we use the constant-acceleration equations we generally say that at some initial time t = 0 an object's position is xo and its velocity is vo. At some other time t its position is x and its velocity is v.
To derive the equations, start with the definition of average acceleration:
| aav |
= |
| Dv
|  |
| Dt
|
|
= |
| v - vo
| |
| t
|
|
If the acceleration is constant then the average acceleration is just the acceleration, so we write:
| a |
= |
| v - vo
| |
| t
|
|
which leads directly to:
Equation 1: v = vo + at
Now let's go back to the definition of average velocity:
| Average velocity: vav |
= |
| Dx
| |
| Dt
|
|
= |
| x - xo
| |
| t
|
|
When the acceleration is constant the average velocity is just the average of the initial velocity vo and the final velocity v, so we get:
| Equation 2 |
| v + vo
| |
| 2
|
|
= |
| x - xo
| |
| t
|
|
That is one ugly looking equation, so we generally use other equations instead.