Question Video: Using Dimensional Analysis to Convert between Units Mathematics

Convert 400 km/h to meters per second, giving your answer correct to one decimal place.

04:47

Video Transcript

Convert 400 kilometres per hour to metres per second, giving your answer correct to one decimal place.

The first thing we wanna do is take 400 kilometres per hour and write it as a fraction, where the numerator is 400 kilometres and the denominator is one hour. In our units we have kilometres in the numerator and hours in the denominator. We need to convert the kilometres to metres and the hours to seconds, so that, in the end, we have some number of metres per one second. To do this, we’ll convert the kilometres to metres first and then convert the hours to seconds. We should remember that one kilometre is 1000 metres. But we need to be really careful about what goes in the numerator and what goes in the denominator.

Right now, we have kilometres in the numerator. And we want to get rid of the kilometres. And that means we’ll have to put the kilometres in the denominator. One kilometre is equal to 1000 metres. Since 1000 metres is equal to one kilometre, we’re multiplying this value by one. We’re not changing its value. We’re only changing the units. What we’ve done is we’ve taken the kilometres in the numerator and the kilometres in the denominator and cancel them out. And so we can say that 400 kilometres per hour will be equal to 400 times 1000 metres per hour. We’ve completed the first step as we now have our units in metres. We could multiply everything now. But since we have more conversions to do, we’ll save that for the final step.

Now, we need to look at our time units, hours. Not everyone memorizes how many seconds are in one hour. But we do know how many minutes are in one hour. One hour equals 60 minutes and one minute equals 60 seconds. We’ll need to use both of these values to make our conversion. This time we have hours in the denominator and we want to get rid of that. So we need to put our hours in the numerator. We can convert from hours to minutes. One hour is 60 minutes. We’ve now taken our hours in the denominator and our hours in the numerator and cancel them out. We currently have metres per minute. Again, we’re multiplying by a value of one. One hour is 60 minutes. And so in our final step, we’ll convert the minutes to seconds. One minute equals 60 seconds.

We have minutes in the denominator, and in the numerator they cancel out. And the units we’re left with are metres per second. The numerator is 400 times 1000 times one times one metres. And the denominator is one times one times 60 times 60 seconds. If we rewrite the numerator to say four times 100000 metres and the denominator I have six times six, which equals 36, and 10 times 10, which equals 100, it makes the simplifying a little bit easier. Four goes into 36 nine times. So we can rewrite four over 36 as one over nine. And we can get rid of the 100 and two zeroes in our numerator. 100000 divided by 100 equals 1000, which leaves us with 1000 metres for every nine seconds.

We need how many metres it would be for one second. To get a one in the denominator, we divide by nine. And if we divide by nine in the denominator, we need to divide by nine In the numerator. 1000 divided by nine equals 111.1 repeating. This would be equal to 111.11111 continuing. To round to one decimal place, we’ll look to the right, to the hundredth place, where there’s also a one. And so we would round down. The one in the tenth place stays the same. And everything to the left stays the same as well. 400 kilometres per hour is the same thing as 111.1 metres per second.

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