In

kinematics, acceleration is the

rate of change of the rate of change of

velocity. The

SI unit used to measure it is -

m s^{-2}

As we know from the laws of indices, that means m/s^{2} (this is just a more convenient way of writing it).

As we said, acceleration is the rate of change of velocity - that means you differentiate the velocity function of an object to get its acceleration function. The displacement function is the first integral of the velocity function, so you need to integrate the acceleration function twice to get the displacement function.

If you draw a graph of velocity against time, then the gradient of this graph represents the acceleration at any given time. If you know the graph's function, then you can differentiate velocity with respect to time to get a function for acceleration. This is necessary when working with variable acceleration, however when working with constant acceleration there are a few simple formulae you can use.

First, we should define our variables. Mathematicians commonly used the following variables in displacement, velocity and acceleration calculations -

`a` = acceleration

`v` - final velocity

`u` initial velocity

`s` - displacement

`t` - time

The first simple relationship we can define for constant acceleration is -

a = v - u
-----
t

Or, acceleration equals change in velocity (because we took away the initial velocity from the final velocity) divided by time.

The next relationship is -

s = ^{1}/_{2}(u + v)t

Or, distance travelled is average velocity multiplied by the time period. Remember, the acceleration is

**constant** - so we are able to use the average velocity safely.

By combining the last two formulae, we get -

s = ut + ^{1}/_{2}at²

This can also be seen easily from a graph of velocity against time, if you sketch one. Displacement is simply the area under this graph. The graph can usually be split into a

rectangular area and a

triangular area - the first part of the above formula calculates the rectangular part of the area, and the second part calculates the triangular area.

Note that if initial velocity (`u`) is zero, then there will be no rectangular area on the graph and `ut` will be zero.

Finally, by combining the last two formulae, we get the relationship -

v² = u² + 2as

Or, the square of the final velocity is equal to the square of the initial velocity, add the product of the acceleration, displacement and 2. This formula is less intuitive, but allows you to do a calculation when you don't know the time variable (this usually only happens in maths lessons).