## Important Kinematics Equations! (Memorize them!)

These equations are known as **equations of motion**. They are only valid if the acceleration is CONSTANT (UNIFORM acceleration).

$$v = u + at \tag{1}$$

$$s = ut + \frac{1}{2} at^{2} \tag{2}$$

$$s = \frac{1}{2} (u + v) t \tag{3}$$

$$v^{2} = u^{2} + 2as \tag{4}$$

,Where

$$v = \text{final velocity}$$

$$u = \text{ initial velocity}$$

$$a = \text{ acceleration}$$

$$t = \text{ change in time}$$

$$s = \text{ change in displacement}$$

**NOTE:** These equations are ONLY valid IF AND ONLY IF acceleration is **constant**. (Very Important!)

## How to select which equation of motion to use?

You select the equations based on the amount of information that you have.

Information that you have | Equations of motion to use | ||||
---|---|---|---|---|---|

s | u | v | a | t | |

$v = u + at$ | |||||

$s = ut + \frac{1}{2} at^{2}$ | |||||

$s = \frac{1}{2} (u + v)$ | |||||

$v^{2} = u^{2} + 2as$ |

If you prefer to read words, the information in the table can be found (in words) below.

$$v = u + at$$

The above equation is suitable to use if you do not have the displacement.

$$s = ut + \frac{1}{2} at^{2}$$

The above equation is suitable to use if you do not have the final velocity.

$$s = \frac{1}{2} (u + v) t$$

The above equation is suitable to use if you do not have the acceleration.

$$v^{2} = u^{2} + 2as$$

The above equation is suitable to use if you do not have the time taken.

## Derivation of Equations of Motion

### Equation 1 – $v = u + at$

Remember that equation for acceleration is the following:

$$a = \frac{v -u}{t}$$

You can re-arrange the equation to give:

$$v = u + at$$

### Equation 2 – $s = \frac{1}{2} (u + v) t$

The velocity of a body moving with uniform acceleration increases steadily. Its average velocity therefore equals half the sum of its initial and final velocities:

$$\text{Average velocity} = \frac{u+v}{2}$$

If $s$ is the distance moved in time $t$, then since $\text{average velocity} = \frac{\text{distance}}{\text{time}} = \frac{s}{t}$,

$$\frac{s}{t} = \frac{u+v}{2}$$

which re-arranges to give:

$$s = \frac{1}{2} (u + v) t$$

### Equation 3 – $s = ut +\frac{1}{2}at^{2}$

Substitute the first equation into the second equation.

You will get the following:

$$s = ut +\frac{1}{2}at^{2}$$

### Equation 4 – $v^{2} = u^{2} + 2as$

Re-arranging the first equation to the following:

$$t = \frac{v-u}{a}$$

Substitute the above equation into the third equation and you will get:

$$v^{2} = u^{2} + 2as$$