# Video: The Relation between Work, Force, and Distance

Which of the following formulas correctly shows the relationship between the magnitude of a force, the work done by that force, and the distance moved in the direction of the applied force by the object that the force is applied to? [A] 𝑊 = 𝑑/𝐹 [B] 𝑊 = 𝐹/𝑑 [C] 𝑊 = 𝑑 − 𝐹 [D] 𝑊 = 𝐹𝑑 [E] 𝑊 = 𝐹 − 𝑑

04:22

### Video Transcript

Which of the following formulas correctly shows the relationship between the magnitude of a force, the work done by that force, and the distance moved in the direction of the applied force by the object that the force is applied to?

Before we review our answer options, let’s consider what’s going on in this situation. We have some object. We’ll say that this is our object. And we imagine that some force is being applied to this object. We can call that force 𝐹. And that along with this, the object moves or is displaced at some distance — we can call that 𝑑 — in the same direction as this force 𝐹 is applied. Based on that, our question is, which of these five options — A, B, C, D, or E — gives us the correct formula, showing the relationship between these two quantities — force, distance — and a third quantity the work done on our object. In other words, we want to know how work, force, and distance are related to one another mathematically.

That being said, let’s look at our answer candidates. A) Work is equal to distance divided by force. B) Work is equal to force divided by distance. C) work is equal to distance minus force. D) Work is equal to force times distance. E) Work is equal to force minus distance.

One of the first things we can do to start narrowing down our answer options is to consider the units involved in these expressions. Each one of these expressions involves three terms. There’s work, 𝑊, force, 𝐹, and distance, 𝑑. Now, if we consider the units of each of these three terms, work, force, and distance, then starting at the top, the unit of work is the joule, abbreviated capital 𝐽. The base unit for force we know to be the newton, abbreviated capital 𝑁. And the base unit for distance or displacement is the meter.

Now, let’s consider for a moment that the unit of work is the joule. Looking at these five answer options, we see that work appears on the left-hand side of each one. And since the unit of work is the joule, that means the left-hand side of every one of these equations has units of joules. Now, if some amount of work in units of joules is equal to some other quantity, that is the quantity on the right-hand side, then that means the right-hand side of the correct formula must also be expressed ultimately in these same units, in units of joules. This means that if we can say for sure that the units on the right-hand side of any of these expressions cannot be joules, in that case, those answer options cannot be the correct formula.

Keeping this in mind, let’s consider answer options C and E. Option C claims that work is equal to distance minus force, where option E claims that work is equal to force minus distance. We know that the units of force are newtons, and the units of distance are meters. But that means that we’re not able to combine the units of these terms in this way and wind up with units of joules, which we would have to, if they were to agree with the left-hand side. For example, for answer option C, we would have a distance in meters minus a force in newtons. But subtracting some number of newtons from some number of meters doesn’t give us some number of joules. The units don’t work out. And this means that answer option C can’t be our correct formula.

Answer option E drops out for the same reason. Some number of newtons minus some number of meters cannot equal some number of joules. So we’re down to answer options A, B, and D. Now, at this point, we can recall that there is a relationship that ties together the units of joules, newtons, and meters. The definition of the unit joule is that one joule is equal to one newton of force multiplied by one meter of distance. Written another way, we can say that a joule is equal to a newton meter. This shows us that whatever our correct answer option is, we’ll have units of newtons multiplied by meters on the right-hand side of our expression.

Looking at option A, we see that this candidate has units of meters divided by newtons, so not newtons times meters, whereas option B has units on the right-hand side of newtons divided by meters. Then, finally, option 𝑑 has units on the right-hand side of newtons multiplied by meters. We see that it’s this option which gives us the units on the right-hand side which agree with the units equivalent to a joule. Therefore, option D has the correct relationship between the units on the left-hand and right-hand side. And therefore, this is our choice for the formula correctly representing the relationship between work, force, and distance. Work is equal to force multiplied by distance.