We are frequently interested in cases where the acceleration is constant. For example, anytime the force comes from gravity it can be assumed to be constant (as long as we are still on Earth, anyway). Therefore it is useful to develop a set of equations to use with constant acceleration problems.
The variables used to describe motion are position (x), velocity (v), acceleration (a), and time (t). What we will do is derive equations that relate just three of these at a time, so that you can use the equation that relates the given quantities with the desired variable, for any particular problem that you may encounter. What follows is a derivation of these formulas, which you may or may not care to follow. At the end is a table of all four formulas.
Because we are talking about an accelerating object, the velocity is obviously not constant. This means that we cannot use the simple formula d=vt anymore because it is only true if v is constant. But it is still true for the average velocity, so we can still say
where the bar over the v indicates average velocity. Because we are only looking at cases where the acceleration is constant, though, we can use the formula for constant acceleration to describe the acceleration at any time:
Also, when the acceleration is constant the velocity increases linearly. What this means is that the average velocity is equal to the velocity at the half-way point (see the picture below), or
Substituting this expression into the formula for the average velocity in equation (1) gives us
Making the substitution v = v0 + at gives
Eliminating t from (3) and (2) gives
Equations for Constant Acceleration
Last Updated 07/08/04 by JWB