## Kinematics

Kinematics is a branch of mechanics which deals with the motion of objects without referencing to the cause of the motion. In kinematics we describe the motion of an object in terms of displacement, distance, speed, velocity and acceleration.

### Displacement vs distance

**Displacement** is the distance between two points in a specified direction. Displacement is a vector quantity, which means that it has both magnitude and direction. To understand the difference between **distance** (which is a scalar quantity) and displacement (which is a vector quantity), lets us consider the following simple example:

- Juliet walks from point A to point B and then back to point B. Given that A and B are 3km apart, the total distance she walked would be 3km+3km = 6km. That is because when dealing with distance we add the magnitudes of the distances, irrespective of the direction in which she was travelling.
- However if asked to find her final displacement, we take into consideration that initially she was travelling towards B and make that displacement positive. As she is then travelling from B back to A the displacement becomes negative because she is now travelling in the opposite direction. Her total displacement becomes 3km-3km = 0km.

The SI unit of distance or displacement is metre (m).

### Velocity vs speed

**Speed** is the rate of change of distance. Speed is a scalar quantity which means that any change in direction of travel won’t affect the speed of an object as long as distance is covered at a constant rate.

**Velocity** is the rate of change of displacement. Velocity is a vector quantity which means that a change in the direction of travel will affect velocity even if the distance is being covered at the same rate.

For example, if a car rounding a bent at a constant speed, the velocity will not be constant because the direction will be changing.

The SI unit of speed or velocity is metre per second (m/s or ms^{-1}).

### Acceleration

**Acceleration** is the rate of change of velocity. Acceleration is a vector quantity.

Take note, if a body is travelling in a certain direction:

- An acceleration in the same direction as the body will increase both its speed and velocity.
- An acceleration in the opposite direction to the direction of travel of the body will decrease both its speed and velocity.
- An acceleration at angle 90° to the direction of travel of the body will change its velocity but will not change its speed.

The SI unit of acceleration is metre per second squared (m/s^{2} or ms^{-2}).

## Kinematics calculations and formulae

From the definition of acceleration, we know that:

$latex acceleration = \frac{change\ in\ velocity}{time\ taken}$

$latex a = \frac{\Delta v}{\Delta t}$

Taking initial velocity to be *u* and final velocity to be *v*, change in velocity = *v*–*u*.

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

Rearranging the formula we get:

$latex v = u+at$ ——-[equation 1]

As we know from the definition of velocity, displacement = velocity × time. However, sometimes the body will be accelerating, it makes sense to use average velocity in the formula. So displacement = average velocity × time and $latex average\ velocity = \frac{v+u}{2}$.

This leads us to:

$latex s = \frac{(v+u)}{2}t$ ———[equation 2]

whereby s is the displacement.

Inserting equation 1 into equation 2 we get:

$latex s = \frac{(u+at+u)}{2}t$

$latex s = \frac{(2u+at)}{2}t$

$latex s = \frac{(2ut+at^2)}{2}$

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

$latex s = ut + \frac{1}{2}at^2$ ——-[equation 3]

Taking equation 1 and making t the subject of formulae we get:

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

Then substitute this equation for t into equation 2:

$latex s = \frac{(v+u)}{2} \times \frac{(v-u)}{a}$

$latex s = \frac{(v+u)(v-u)}{2a}$

$latex s = \frac{v^2-u^2}{2a}$

$latex 2as = v^2-u^2$

$latex v^2 = u^2+2as$ ———–[equation 4]

These equations of motion can only be used if there is uniform acceleration.