In this explainer, we will learn how to use different units of speed and to convert between them.
The SI unit of speed is the metres per second (m/s).
Speed is not always expressed in metres per second, however.
For example, when referring to small objects moving on tabletops, such as in practical experiments, the unit of centimetre per second (cm/s) is sometimes used.
Distances in metres and centimetres can be converted to and from each other by recalling that the product of the value and unit of measurement must be equal for a measurement whatever unit is used.
There are 100 centimetres in a metre. A distance can then be converted between metres and centimetres as follows:
Units of speed in which only the unit of distance is changed, such as centimetres per second (cm/s), millimetres per second (mm/s), and kilometres per second (km/s), require only that the value of the speed be multiplied by the conversion factor that relates the units of distance involved in the conversion.
Let us now look at such an example.
Example 1: Converting from Kilometers per Second to Meters per Second
A spacecraft travels at a speed of 5.5 kilometres per second. What is its speed in metres per second?
Answer
The unit “kilometres per second” means and has the symbol “km/s.”
The units “km/s” and “m/s” both consist of a distance divided by a time in seconds.
The value of a speed expressed in metres per second that is equivalent to a value expressed in kilometres per second is found by converting the value of the distance traveled expressed in kilometres to the same distance expressed in metres.
A speed of 5.5 km/s can be written as
The prefix “kilo” means “1 000,” so the conversion factor from kilometres to metres has the value 1 000:
We see then that a speed of 5.5 km/s is equivalent to a speed of
Units of speed do not always consist of a distance divided by a time measured in seconds. Other units of time can be used.
A unit often used for speed that does not measure time in seconds is the unit “kilometres per hour,” which is written as kph, where “p” represents “per.” This unit can also be written as km/h, which it often is.
These units are convenient units for vehicle users, as the distances of travel taken in such vehicles are typically much better known by their users in kilometres than in metres, and the associated travel times are much better known in hours than in seconds. The familiarity to many people of kilometres per hour as a unit of speed can be useful in easily determining whether a calculation of the speed of an object is likely to be correct if the speed at which such an object moves in kilometres per hour is familiar.
For example, someone attempts to calculate the speed of a car in metres per second. They obtain an answer of 90 m/s.
It may not be obvious to the person whether this is a realistic speed for a car. It is not necessarily clear whether 90 m/s is slow, or fast, or very fast, or ridiculously fast to a person who is not familiar with the speeds of cars expressed in metres per second.
A speed of 90 m/s is in fact equivalent to a speed of 324 kph. This is easily recognizable as much faster than most cars are capable of moving.
When converting a speed in metres per second to a speed in a unit in which both the unit of distance and the unit of time are changed, the change to each unit must be considered.
A distance of 90 m is converted to 90 km by division by 1 000 as follows:
We see then that
The conversion between seconds and hours depends on the fact that and
From this, we see that
In other words, 1 hour equals 3 600 seconds.
To convert a speed of 0.09 km/s to a speed in kilometres per hour, the value of time in seconds must be converted to a value of time in hours.
The conversion factor of seconds to hours is shown below:
Substituting the value of the hours that is equivalent to 1 second, we have that
This expression now contains a fractional term:
It is the case that
A factor of in the denominator of the fractional term is equivalent to a factor of 3 600 in the numerator in the term, and so the expression for the speed can be written as
We can now separate the value terms (the numeric ones) from the units, giving us
This gives us a speed of 324 kph.
Let us now look at an example involving a conversion between units of speed in which both the time unit and the distance unit are changed.
Example 2: Converting from Meters per Second to Kilometers per Hour
What value of speed in kilometres per hour is equal to a value of speed of one metre per second?
Answer
We start with a speed expressed in metres per second. We want to express this speed in kilometres per hour (kph, which is the same thing as km/h).
We know that the conversion factor from metres to kilometres is :
We know that the conversion factor from seconds to hours is :
We see then that
We can separate the value terms in the right side of the expression from the unit terms. This gives us
It is the case that
We see then that
A speed of 1 m/s is equivalent to a speed of 3.6 kph.
Let us now look at an example involving the reverse operation to the previous example.
Example 3: Converting from Kilometers per Hour to Meters per Second
What value of speed in metres per second is equal to a speed of one kilometre per hour? Answer to three decimal places.
Answer
We start with a speed expressed in kilometres per hour (kph, which is the same thing as km/h). We want to express this speed in metres per second.
We know that the conversion factor from kilometres to metres is 1 000:
We know that the conversion factor from hours to seconds is 3 600:
We see then that
We can separate the value terms in the right side of the expression from the unit terms. This gives us
A speed of 1 km/h is equivalent to a speed of
To three decimal places, this is 0.2778 m/s.
As well as the unit “kilometres per hour,” the unit “miles per hour,” is another commonly used unit of speed. This unit is usually only written as mph and not as m/h.
The conversion factor between miles and kilometres has an approximate value given by
Miles is written as “mi” so as not to confuse miles with metres.
Let us now look at an example involving conversion from a speed expressed in metres per second to the same speed expressed in miles per hour.
Example 4: Converting from Miles per Hour to Meters per Second
What value of speed in metres per second is equal to a speed of 1 mile per hour? Round your answer to two decimal places.
Answer
We start with a speed expressed in miles per hour. We want to express this speed in metres per second.
One way to solve this problem is to first convert the distance unit from miles to kilometres.
A distance of 1 mile equals a distance of 1.6 kilometres.
We know that the conversion factor from kilometres to metres is 1 000.
We know that the conversion factor from hours to seconds is 3 600.
We see then that, for a speed of 1 mph,
We can separate the value terms in the right side of the expression from the unit terms. This gives us
A speed of 1 mph is equivalent to a speed of
To two decimal places, this is 0.44 m/s.
Finally, let us look at an example in which the distance unit of a speed is not changed but the time unit is.
Example 5: Converting from Meters per Second to Meters per Hour
A snail travels at a speed of 0.0025 metres per second. What is its speed in metres per hour?
Answer
The only conversion needed in this example is from seconds to hours. We know that 1 second is hours.
We see then that
The term is equivalent to
We have then that
It is helpful to notice that 9 metres is a realistic distance for a snail to travel in an hour. If the result obtained had been nearer to 100 metres than to 10 metres, that would be a good reason to think that an error had occurred.
Let us now summarize what has been learned in these examples.
Key Points
- Speeds are sometimes expressed in kilometres per hour or in miles per hour.
- If a speed expressed in metres per second is converted into a speed expressed using a unit other than metres for distance and where seconds is used as the unit of time, the conversion factor between the speeds is the conversion factor between metres and the other distance unit.
- If a speed expressed in metres per second is converted into a speed expressed as a distance in metres per time unit other than seconds, the conversion factor between the speeds is the conversion factor between the other time unit and seconds.
- If a speed expressed in metres per second is converted into a speed expressed using a unit other than metres for distance and a unit other than seconds for time, the conversion factor between the speeds is the ratio of the conversion factor between metres and the other distance unit to the conversion factor between seconds and the other time unit.
- 1 mile is approximately 1.6 kilometres. This conversion factor is often taken as exact rather than approximate.
