## Unit Conversions

Often a quantity such a distance is given in one set of units when it would be better to have in a different set of units. In such cases it is important that you are able to convert such that the answer has the correct units.

For example, suppose you are driving a car at 30 mph for 15 minutes. You can calculate the distance travelled from the formula distance = speed x time, but are faced with the problem that the speed is measured in miles per hour, whereas the time is measured in minutes. However, if you remember that there are 60 minutes in an hour, you can convert the time to be 0.25 hours, in which case the distance can be calculated as

Distance = 30 mph x 0.25 h = 7.5 miles.

In astronomy distances can be very large, and are often expressed in units which allow numerical values to be kept reasonably small, but which are unfamiliar to you. For example, the Andromeda galaxy is 2 million light years away. How far is that really? What is a light year? What is the distance in miles or in kilometers, units with which you are more familiar?

In trying to answer this question you need to know how to convert units reliably, even when the quantities involved are outside your normal experience. The method is based on that used in the simple example above. In converting the time from minutes to hours you multiplied 15 minutes by a conversion factor to get 0.25 hours. The trick is to make sure you get the conversion factor the right way up.

#### Conversion method

Suppose I want to know how far 5 A.U. is in metres. In a table of conversion factors I find one A.U. is equal to 1.5 x 1011 m. The ratio of these numbers is therefore just one. If I take my distance of 5 A.U. and multiply by this ratio, multiplying by one does not change the value. All I have to choose know is whether 1 A.U. goes on the top of the ratio and 1.5 x 1011 m on the bottom, or 1.5 x 1011 m on the top and 1 A.U. on the bottom. If I look at the units I can make that decision easily. My original distance is in A.U., units which I want to discard, so if I write my ratio with A.U. on the bottom, then the unwanted units will 'cancel', leaving me with the units of metres, which I want.

#### Another example

What is 400 A.U. expresssed in miles? What is it expressed in light years?

Since we know from the conversion table that there are 93 million miles in one A.U. then the conversion from A.U. to miles is

Now that we have the answer in miles, we can use the conversion factor that there are 6 trillion (6 x 1012) miles in a light year to convert this distance into light years

#### Multiple conversion factors

Sometimes you cannot make the conversion in one step because you do not know the conversion factor. However you may be able to make the conversion using one or more intermediate steps. For example, Venus goes around the Sun in 225 (Earth) days. How long is this in seconds? You probably do not know how many seconds there are in a day (the direct conversion factor) but you do know how many there are in a minute, how many minutes there are in an hour, and how many hours in a day. Putting this all together

#### Mixed units

You can also use this technique to convert two or more units in the same expression. For example, convert 30 m/s to miles per hour. You need to convert metres to miles and seconds to hours.

Note that the conversion could have been performed in two separate conversions, but can also be done using two multiplying factors in the one line, the first to make the required conversion metres to miles, and the other to convert seconds to hours.

#### An Astronomical Example

The Andromeda Galaxy is 2 million light years away. If you have a spacecraft capable of travelling at one million miles an hour, how long would it take you to get there?

The calculation is a simple one, the time being calculated from the formula time = distance /speed. Note that since the speed is quoted in miles per hour, and the distance is in light years, one will need to be converted. The answer will then be expressed in hours (since that was the time unit in the speed) and should be converted to years to give a true idea of just how long it takes.

More than a billion years. This gives us an idea of how vast the universe is, and the immense times it takes to cover even a small portion of it. The idea portrayed in science fiction programs such as Star Trek that inter-galactic travel is possible within a one hour episode is not feasible. (You could argue that with a faster speed you could travel the same distance in less time, which is true. However, even with the best possible technology, you can never hope to travel faster than the speed of light, and the journey to the Andromeda Galaxy will take at least 2 million years.)

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