# Equations of Motion

Show/Hide Sub-topics (Speed, Velocity and Acceleration | O Level)
Show/Hide Sub-topics (Kinematics | A Level)

## 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.

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:

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