Speed

Definition of Speed

Speed is a measure of how fast an object moves. It is defined as the distance travelled divided by the time taken to travel that distance.

Mathematically, speed is given by the formula:

$$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$

Speed is a scalar quantity, which means it only has magnitude (size) and no direction.

Common units of speed include:

• Metres per second ($\mathrm{m\,s^{-1}}$)
• Kilometres per hour ($\mathrm{km\,h^{-1}}$)

For example, if a car travels 100 metres in 20 seconds, its speed is:

$$\text{Speed} = \frac{100\,\mathrm{m}}{20\,\mathrm{s}} = 5\,\mathrm{m\,s^{-1}}$$

For instance, if a person walks 3 km in 1 hour, their speed is $3\,\mathrm{km\,h^{-1}}$.

Calculating Speed

To calculate speed, use the formula:

$$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$

Make sure the units of distance and time are compatible before calculating speed. For example, if distance is in kilometres and time in hours, speed will be in $\mathrm{km\,h^{-1}}$. If distance is in metres and time in seconds, speed will be in $\mathrm{m\,s^{-1}}$.

If units are mixed, convert them first (see the next section on unit conversion).

For example, if a cyclist travels 15 kilometres in 30 minutes, first convert 30 minutes to hours:

30 minutes = $\frac{30}{60} = 0.5$ hours

Then calculate speed:

$$\text{Speed} = \frac{15\,\mathrm{km}}{0.5\,\mathrm{h}} = 30\,\mathrm{km\,h^{-1}}$$

Average Speed

When an object moves at different speeds during a journey, its average speed is the total distance travelled divided by the total time taken.

The formula for average speed is:

$$\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}$$

This is useful in real life when speeds vary, such as in traffic or during a race with varying pace.

For example, if a car travels 60 kilometres in 1 hour and then 90 kilometres in 2 hours, the total distance is 150 kilometres and the total time is 3 hours, so the average speed is:

$$\frac{150\,\mathrm{km}}{3\,\mathrm{h}} = 50\,\mathrm{km\,h^{-1}}$$

Speed and Unit Conversion

Often, you need to convert between different units of speed, especially between $\mathrm{km\,h^{-1}}$ and $\mathrm{m\,s^{-1}}$.

To convert from $\mathrm{km\,h^{-1}}$ to $\mathrm{m\,s^{-1}}$:

Multiply by $\frac{1000}{3600} = \frac{5}{18}$

So,

$$\text{Speed in } \mathrm{m\,s^{-1}} = \text{Speed in } \mathrm{km\,h^{-1}} \times \frac{5}{18}$$

To convert from $\mathrm{m\,s^{-1}}$ to $\mathrm{km\,h^{-1}}$:

Multiply by $\frac{3600}{1000} = \frac{18}{5}$

So,

$$\text{Speed in } \mathrm{km\,h^{-1}} = \text{Speed in } \mathrm{m\,s^{-1}} \times \frac{18}{5}$$

Also, when converting time units, remember:

• 1 hour = 3600 seconds
• 1 minute = 60 seconds

For example, to convert 54 $\mathrm{km\,h^{-1}}$ to $\mathrm{m\,s^{-1}}$:

$$54 \times \frac{5}{18} = 54 \times \frac{5}{18} = 15\,\mathrm{m\,s^{-1}}$$