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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/s2 (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 + 1/2at²

```
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).

Ac*cel`er*a"tion (#), n. [L. acceleratio: cf. F. acc'el'eration.]

The act of accelerating, or the state of being accelerated; increase of motion or action; as, a falling body moves toward the earth with an acceleration of velocity; -- opposed to retardation.

A period of social improvement, or of intellectual advancement, contains within itself a principle of acceleration. I. Taylor.

Astr. & Physics. Acceleration of the moon, the increase of the moon's mean motion in its orbit, in consequence of which its period of revolution is now shorter than in ancient times. -- Acceleration and retardation of the tides. See Priming of the tides, under Priming. -- Diurnal acceleration of the fixed stars, the amount by which their apparent diurnal motion exceeds that of the sun, in consequence of which they daily come to the meridian of any place about three minutes fifty-six seconds of solar time earlier than on the day preceding. -- Acceleration of the planets, the increasing velocity of their motion, in proceeding from the apogee to the perigee of their orbits.

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