# Video: Combining Quantities with Different Units

Which of the following is equivalent to (3kg × 3 m/s² × 3 m)/3 s? [A] 9 J [B] 9 N ⋅ m [C] 9 W [D] 9 N [E] 9 kg ⋅ m/s

06:10

### Video Transcript

Which of the following is equivalent to three kilograms times three metres per second squared times three metres divided by three seconds? a) Nine joules, b) Nine newton metres, c) Nine watts, d) Nine newtons, e) Nine kilogram metres per second.

The question is asking us to reduce this large expression into one of these five, more compact expressions. Focusing on the large expression in the question, we know that every term involved is a term with units. So, to reduce this expression, we’ll need some ground rules for working with terms with units. We’ll start with a definition. A term with units consists of a number multiplying a unit, where a unit is just what we use to measure some quantity.

So, for example, in a physics problem, we might use kilograms as a unit for mass or metres per second as a unit for speed. On the other hand, if we lived on an apple orchard, we might use barrels of apples per day as a unit for productivity. The important thing about a unit is that it identifies what we’re counting. And a number, of course, tells us how many of them we’ve counted. Then, a term with units, which is a number times a unit, answers the questions, what are we counting? And how many of it have we counted?

With this definition in hand, we need a rule for multiplying and dividing terms with units, so that we can reduce the expression in the question. The rule is straightforward. Treating the number and unit parts separately, multiply or divide the numbers, multiply or divide the units. The resulting number and resulting unit, then, become the number and unit parts of the resulting term.

As an example of how to apply this rule, let’s reduce five metres times two newtons divided by one second. First, we’ll multiply and divide the numbers while leaving the units alone. We have five times two divided by one. Five times two divided by one is 10 divided by one, which is just 10.

Now, let’s turn to the units and leave the numbers alone. We have metres times newtons divided by seconds. Metres times newtons divided by seconds is newton metres per second. Now, combining our resulting number and our resulting unit, we find that five metres times two newtons divided by one second is equivalent to 10 newton metres per second. This also illustrates the general result that when multiplying or dividing terms with units, the final unit is different from the individual starting units.

Now that we can multiply and divide terms with units, let’s apply this rule to the expression in the question. Again, starting with the numbers, we have three times three times three divided by three. Three times three times three is 27 divided by three equals nine. So, our resulting number for a reduced expression is nine.

Now, we’ll do the same for the units. We have kilograms times metres per second squared times metres divided by seconds. Metres per second squared times metres is metres squared per second squared. Kilograms times metres squared per second squared is kilograms metres squared per second squared. And kilograms metres squared per second squared divided by seconds is kilograms metres squared per second cubed, which is the resulting unit for our reduced expression.

Putting our resulting number and resulting unit together, we find that the large expression in the question can be reduced to nine kilograms metres squared per second cubed. Alright, now we just need to compare this expression to our five answer choices. And the correct choice will have both a number and a unit that agrees with our number and our unit. We see that all the numbers agree because they’re all nine. So, that leaves us to find the answer choice whose unit agrees with kilograms metres squared per second cubed.

The trouble is that answer choices a, b, c, and d all contain units that aren’t kilograms, metres, or seconds. So, we’ll need to convert these units into kilograms, metres, and seconds to compare them to our unit. The units we need to convert are joules, newtons, and watts. Joules are equivalent to kilogram metre squared per second squared. Newtons are equivalent to kilogram metres per second squared. And watts are equivalent to kilogram metres squared per second cubed. We can now replace the joules, newtons, and watts in choices a through d with kilograms, metres, and seconds.

Note that choice b has unit of newton metres, so the equivalent unit is the equivalent of newtons with an extra factor of metres. Written out this way, we see immediately that choice c, nine watts, has number and unit that match the expression in the question. We found the answer by simply writing down the conversions between joules, newtons, and watts and kilograms, metres, and seconds. But if we had forgotten some of those conversions, we could’ve worked them out by recalling that the units on both sides of a physics equation must agree.

Say we wanted to work out the conversion for newtons. We start by recalling that newtons measure force. Now, we take an equation for force like Newton’s second law, force equals mass times acceleration. The units of force are newtons, the units of mass are kilograms, and the units of acceleration are metres per second squared. And since the units on both sides must agree, we see that newtons are equivalent to kilograms metres per second squared. And so, we’ve worked out for ourselves the conversion between newtons and kilograms, metres, and seconds.

Let’s do the same for joules. Joules measure energy. Perhaps we remember that energy is equal to force times distance. Or perhaps, you remember that kinetic energy is one-half mass times speed squared. In either case, the units of joules on the left must be equivalent to the units on the right. For force times distance, that’s newton metres, and for one-half mass times speed squared, that’s kilograms metres squared per second squared since the unit of speed is metres per second.

Note that using force times distance got us newton metres, which isn’t quite where we wanted to be. We would still have to further convert newtons, either by knowing the conversion or rederiving as we did a moment ago. Finally, just for completeness, watts measure power. And a simple equation for power is that power is energy per time.

Power and watts, energy and joules, and time and seconds. So, a watt is a joule per second. Again, converting joules further as necessary. So, we see that even if we’ve forgotten a unit conversion, we can still work that conversion out from relevant physics equations and get the answer.