The Equations for Uniform Acceleration

There are four equations for uniform acceleration (also known as the kinematic equations) which are used to describe the motion of an object:

1.
2.
3.
4.

Where:
u – initial velocity
v – final velocity
a – acceleration
t – time
s – displacement


You may be wondering – why there are four equations instead of one? Notice that each equation has one variable missing. For example, the first equation doesn’t have the variable displacement, s, in it. So, if you were to calculate something that involves displacement, equation 1 is not your choice. Basically, if you know any three of u, v, a, t and s, the others can be found using one or more of the above equations. But take note that you can use these equations only when the acceleration of an object is constant throughout its motion.

It’ll be useful to memorize these formulas if you’re going to solve problems involving motion, but do you know how to derive them? Knowing how to derive these formulas is useful because you will understand how these equations originated and this knowledge acts as a backup in case you forgot any one of them – you just have to derive it and they just appear out of nowhere!


Deriving the Equations

Suppose the velocity of a body increases at a consistent rate from u to v in time t, the body is said to be accelerating uniformly and uniform acceleration a is given by


Since the velocity is increasing steadily, the average velocity is the mean of the initial and final velocities:


If s is the displacement of the body in time t, then the average velocity is equal displacement/time or s/t, so we can say


By substituting equation 1 into equation 2, we have


By rearranging equation 1, we know that t = (v-u)/a. We substitute this value of t into equation 2 and we have


There’re other ways of deriving these equations. Can you find them?


See Also

• Problem: Kinematics #1 - a sample problem that requires the usage of these equations.