Speed and velocity

How fast?

Two buses go to the next stop. One takes $5\ \text{min}$, the other $8\ \text{min}$. The same distance, different times. Speed is the distance covered in one unit of time:

In science we use metres per second (m/s); on roads, kilometres per hour (km/h).

From km/h to m/s

One kilometre is $1000\ \text{m}$ and one hour is $3600\ \text{s}$, so $1\ \text{km/h} = \tfrac{5}{18}\ \text{m/s}$.

Average speed

Real trips speed up, slow down and stop. For a whole trip we use

It is the steady speed that would cover the same distance in the same time.

You should find that the average speed is usually not the mean of the two speeds. The slower part takes longer, so it pulls the average down. They are equal only when the two parts take the same time.

Velocity

Speed says how fast. Velocity says how fast and in which direction:

A runner does one lap of a $400\ \text{m}$ track in $80\ \text{s}$. Her average speed is $5\ \text{m/s}$, but her displacement is zero, so her average velocity is zero.

A vehicle's odometer shows the distance it has travelled; its speedometer shows its speed at this moment.

Watch out. Motion is uniform only if equal distances are covered in equal times, however small the times. Most real motion is non-uniform.

A worked example

A scooter goes $12\ \text{km}$ at $36\ \text{km/h}$, then $8\ \text{km}$ at $16\ \text{km/h}$. Find its average speed.

1. Times. $\dfrac{12\ \text{km}}{36\ \text{km/h}} = \tfrac13\ \text{h} = 20\ \text{min}$ and $\dfrac{8\ \text{km}}{16\ \text{km/h}} = \tfrac12\ \text{h} = 30\ \text{min}$.
2. Average speed. $\dfrac{20\ \text{km}}{50\ \text{min}} = \dfrac{20\ \text{km}}{\tfrac56\ \text{h}} = 24\ \text{km/h}$.
3. So the average, $24\ \text{km/h}$, is less than the mean of the speeds, $26\ \text{km/h}$.

Practise with new numbers

Try it in Python

The program works out the scooter trip.

Remember these

• Speed $=$ distance $\div$ time; velocity $=$ displacement $\div$ time.
• $1\ \text{km/h} = \tfrac{5}{18}\ \text{m/s}$.
• Average speed $=$ total distance $\div$ total time, not the mean of the speeds.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A cyclist covers $90\ \text{m}$ in $15\ \text{s}$. What is her speed?

2. What is $54\ \text{km/h}$ in m/s?

3. A car goes $150\ \text{km}$ in $3$ hours, with a stop for lunch. What is its average speed?

4. Can an object have a large average speed but zero average velocity?