Velocity

Definition of Velocity

Velocity is a vector quantity, which means it has both a magnitude (size) and a direction. This distinguishes it from speed, which is a scalar quantity and only tells you how fast something is moving without any information about direction.

For example, if a car is travelling at 20 m/s to the north, its velocity is 20 m/s north. If it changes direction but keeps the same speed, its velocity changes because the direction part of the vector changes.

Calculating Velocity

Velocity is calculated by dividing displacement by time. Displacement is the straight-line distance from the starting point to the finishing point, including direction.

The formula is:

$\text{Velocity} = \frac{\text{Displacement}}{\text{Time}}$

Velocity is measured in metres per second (m/s).

Because velocity is a vector, displacement must be used (not distance), so direction is important. For example, if a runner goes 100 m east in 20 seconds, the velocity is:

$\text{Velocity} = \frac{100 \text{ m east}}{20 \text{ s}} = 5 \text{ m/s east}$

Velocity-Time Graphs

A velocity-time graph shows how velocity changes over time.

• The gradient (slope) of the graph represents acceleration. A positive gradient means speeding up, a negative gradient means slowing down, and a zero gradient means constant velocity.
• The area under the graph represents displacement during that time period.

For example, a flat horizontal line means the velocity is constant. A straight line sloping upwards means the velocity is increasing steadily.

If the velocity-time graph crosses the time axis (velocity = 0), it means the object has changed direction.

For instance, if a velocity-time graph shows a velocity of 5 m/s for 10 seconds, the displacement is:

$\text{Displacement} = \text{velocity} \times \text{time} = 5 \times 10 = 50 \text{ m}$

Instantaneous and Average Velocity

Instantaneous velocity is the velocity of an object at a specific instant in time. It can be found by looking at the velocity at a single point on a velocity-time graph.

Average velocity is the total displacement divided by the total time taken. It is useful when the velocity is not constant (non-uniform motion).

The formula for average velocity is:

$\text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time}}$

For example, if a cyclist travels 200 m north in 50 seconds, then 100 m south in 30 seconds, the total displacement is 100 m north (200 m north - 100 m south) and the total time is 80 seconds. So the average velocity is:

$\frac{100 \text{ m north}}{80 \text{ s}} = 1.25 \text{ m/s north}$

Example: A car moves 150 m east in 30 seconds, then 50 m west in 10 seconds. Calculate the average velocity.

Total displacement = 150 m east - 50 m west = 100 m east

Total time = 30 s + 10 s = 40 s

Average velocity = $\frac{100 \text{ m east}}{40 \text{ s}} = 2.5 \text{ m/s east}$