Video: Power

Which of the following formulas correctly shows the relationship between the time taken to do an amount of work and the power supplied to do the work? [A] 𝑃 = 𝑡/𝑊 [B] 𝑃 = 𝑊𝑡 [C] 𝑃 = 𝑊 − 𝑡 [D] 𝑃 = 𝑊/𝑡 [E] 𝑃 = 𝑊 + 𝑡

03:42

Video Transcript

Which of the following formulas correctly shows the relationship between the time taken to do an amount of work and the power supplied to do the work? A) Power equals time divided by work. B) Power equals work times time. C) Power equals work minus time. D) Power equals work divided by time. E) Power equals work plus time.

We’re told that these five options are all candidates for the correct formula showing the relationship between the time taken to do an amount of work and the power supplied to do it. We’re working then with these three particular quantities, power, work, and time. Now even if we don’t have the exact relationship between these three terms committed to memory, if we’re able to recall the units of these three terms, we’ll be able to narrow down our answer options.

Starting from the bottom, the SI base unit of time is the second, abbreviated s. The unit for work is the joule, abbreviated capital J. And, notice, that this is the same as the unit for energy. And then, lastly, the SI unit for power is the watt, abbreviated capital W.

As we consider our answer options, we know that whichever one is correct will have the same units on either side of the equality sign. That’s part of what makes it an equation. And since we know that the SI unit of power is the watt, we realize that all five of our options will have that unit on the left-hand side of the equality, which means that whichever answer option is correct must have that same unit, the watt, on the right-hand side.

As we consider answer options C and E, we see that the right-hand side will not have those units for these two choices. For these two choices, what we have on the right-hand side is a mixture of units. The work has units of joules and the time has units of seconds. In each of these cases then, we’re not able to perform the operation on the right-hand side. We’re not comparing quantities of a similar type to one another. From a units perspective then, we can say that options C and E are out of the running. The final unit on the right-hand side of these expressions can’t possibly be watts. And so, it can’t be a correct formula.

That leaves A, B, and D. At this point, it’ll be helpful to us to recall the definition of power. Power is defined as energy transferred per unit time. In other words, it’s an amount of energy in units of joules divided by an amount of time in units of seconds. And note that this energy transfer can be happening thanks to the work done by some entity. We could equally well define power as work done per unit time.

So, the unit of work, as we saw, is the joule. And the base unit of time is the second. And our definition for power is telling us that power is an amount of energy in joules per amount of time in seconds. To answer our question then, we’ll go through answer options A, B, and D and see which one has units of joules per second on the right-hand side.

Let’s start out by testing answer option A. On the right-hand side of this equation, we have a time in units of seconds divided by work, which has units of joules. That’s not an overall unit of joules per second. So, we’ll cross out option A. Moving on to option B, this right-hand side has work done in units of joules multiplied by time in units of seconds. The unit here is a joule times second rather than a joule per second, so option B is also off the table.

Lastly, we get to option D, where we have work done in joules divided by time in seconds. Here, we have a match for the units we were looking for, a match for the units of power. We can say then that a joule per second is equal to a watt, the unit of power. And this makes option D the correct choice. Power is equal to work divided by time.

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