How to Safely Convert Between Units
Let's start with an example:
90 kilometers per hour to meters per second
A kilometer has 1000 meters, and an hour has 3600 seconds, so:
How did I know to make it 10003600 and not 36001000 (the other way around)?
The trick is to do the conversions as fractions!
Example: convert 3 km to m (3 kilometers to meters)
There are 1000 m in 1 km, so the conversion is easy, but let's follow a system.
The system is:
- Write the conversion as a fraction that equals 1
- Multiply it out (leaving all units in the answer)
- Cancel any units that are both top and bottom
We can write the conversion as a fraction that equals 1:
1000 m1 km = 1
And it is safe to multiply by 1 (does not affect the answer):
3 km × 1 = 3 km
so we can do this:
3 km × 1000 m 1 km = 3000 km · m 1 km
The answer looks strange! But we aren't finished yet ... we can "cancel" any units that are both top and bottom:
3000 km · m 1 km = 3000 m
So, 3 km equals 3000 m. Well we knew that, but we want to follow a system, so that when things get harder we know what to do!
And when we do it correctly we get to cancel units that are both top and bottom, and get a neat answer.
Note: if we do it wrong (with the conversion upside down) we get this:
3 km × 1 km 1000 m = 3 km · km 1000 m
And that doesn't let us do any cancelling!
Example: km/h to m/s (kilometers per hour to meters per second)
We will do it in two stages:
- from km/h (kilometers per hour) to m/h (meters per hour), then
- from m/h (meters per hour) to m/s (meters per second).
1. From km/h (kilometers per hour) to m/h (meters per hour)
1 km h × 1000 m1 km = 1000 km · m 1 h · km
Now "cancel out" any units that are both top and bottom:
1000 km · m 1 h · km = 1000 m 1 h
It is now in meters per hour.
2. From m/h (meters per hour) to m/s (meters per second)
To go from m/h (meters per hour) to m/s (meters per second) we put the "3600 seconds in an hour" conversion "upside down" because we want an "h" on top (so they will cancel later) :
1000 m 1 h × 1 h 3600 s = 1000 m · h 3600 h · s
Then "cancel out" any units that are both top and bottom:
1000 m · h 3600 h · s = 1000 m 3600 s
And so our anwer is:
1000 m 3600 s = 0.2777... m/s
Doing it wrong (with the the 3600 seconds/hour the other way around) gets this:
1000 m 1 h × 3600 s 1 h = 1000 × 3600 m · s 1 h · h
And there is nothing to cancel!
So we know we made a mistake, and can correct it.
All In One Go
With experience we can do it in one line like this:
1 km 1 h x 1000 m 1 km × 1 h 3600 s
= 1000 km · m · h 3600 h · km · s
= 1000 m 3600 s
Or even "all in one go" (crossing out as we go) like this:
1 km 1 h x 1000 m 1 km × 1 h 3600 s = 1000 m 3600 s
Example: What is 60 mph (miles per hour) in m/s (meters per second) ?
There are 1609 meters in a mile, and 3600 s in an hour
60 mile h × 1609 m mile × 1 h 3600 s
= 60 × 1609 mile · m · h 3600 h · mile · s
= 26.82 m/s
Summary
The important points are:
- Write the conversion as a fraction (that equals one)
- Multiply it out (leaving all units in the answer)
- Cancel any units that are both top and bottom