Video: Knowing the Relationship between Time, Distance, and Speed

Which of the following formulas correctly relates time, distance, and speed? [A] 𝑠 = 1/𝑑𝑑 [B] 𝑑 = 𝑠 βˆ’ 𝑑 [C] 𝑠 = 𝑑 + 𝑑 [D] 𝑠 = 𝑑𝑑 [E] 𝑠 = 𝑑/𝑑

08:23

Video Transcript

Which of the following formulas correctly relates time, distance, and speed? (A) 𝑠 equals one over 𝑑𝑑, (B) 𝑑 equals 𝑠 minus 𝑑, (C) 𝑠 equals 𝑑 plus 𝑑, (D) 𝑠 equals 𝑑 multiplied by 𝑑, (E) 𝑠 equals 𝑑 over 𝑑.

Okay, so in this question, we’re presented with five different possible formulas and asked which of them correctly relates the three quantities time, distance, and speed. We have time, which is labeled 𝑑; distance, which is labeled 𝑑; and speed, which is labeled 𝑠. Now, whenever we’re trying to work out if a formula is correct, it can be helpful to look at the units. In order for a particular formula to be correct, the units of the left-hand side of that formula must equal the units of the right-hand side. In other words, if the left- and right-hand sides of a formula have different units to each other, then we know that that formula cannot be correct.

To see why this is, remember that the equal sign in an equation means that the left-hand side is equal to the right-hand side. If those two sides have different units, we cannot compare them in a meaningful way to say that they are equal. For example, we could say that three apples is equal to two apples plus one apple. In this case, the units on both sides of the equation are units of apples. So the units on both sides of this equation agree with each other, and this equation makes sense.

But if instead, on the left-hand side, we had three oranges, then we now have an equation where the right-hand side has units of apples and the left-hand side has different units, units of oranges. And it should be clear that we cannot compare a quantity in units of oranges to a quantity in units of apples. So this equation makes no sense.

Since we cannot compare quantities with different units in this way, this also means that we cannot add or subtract two quantities with different units to each other. If we go back to our analogy with fruit, we can add two apples to one apple and get a result in units of apples, in this case, three of them. But if we try to add two apples to one orange, then we see that this doesn’t make any sense. We don’t know how to add together these two quantities because they have different units.

So we’ve now identified a couple of things that we can check with the units of our quantities in order to see if a particular formula could be correct. So now let’s work through our list of potential formulas relating time, distance, and speed that were given in the question to see if any of them fulfill the necessary requirements on the units of the quantities in order to be correct.

First off, we need to identify the units of each of the quantities involved in these equations. If we work in SI units, then we have that the quantity time, if we work in SI units, then we have that the quantity time with symbol 𝑑 has units of seconds. We have that distance with a symbol 𝑑 has units of meters. And finally, we have that speed with a symbol 𝑠 has units of meters per second.

Okay, looking at the potential formulas that we are given, let’s start with option (A). This formula says that 𝑠 is equal to one divided by 𝑑 times 𝑑. So does this formula make sense in terms of the units? If we look at the left-hand side, we have speed. And if we look in our table, we see that speed has units of meters per second. Now looking at the right-hand side of the formula, we have one divided by distance, which has units of meters, and time, which has units of seconds. And so we have that the units are one divided by units of meters multiplied by units of seconds.

So can this formula be correct? Well, if we write this a little more clearly, we have that the left-hand side has units of meters divided by seconds, while the right-hand side has units of one divided by meters multiplied by seconds. And since the units on the left-hand side do not equal the units on the right-hand side, then we have not met our first requirement on the units of the formula. And so we have that option (A) cannot be correct.

Now let’s look at option (B). This formula says that distance 𝑑 is equal to speed 𝑠 minus time 𝑑. If we look at the right-hand side of this formula in terms of the units, we see that we have speed, a quantity with units of meters per second, minus time, a quantity with units of seconds. And so we are trying to subtract one quantity from another when those two quantities have different units. So this contradicts our second requirement for the units of a formula. And so option (B) cannot be correct.

Now, let’s look at option (C). We see that this formula is telling us that speed 𝑠 is equal to distance 𝑑 plus time 𝑑. And if we look at the units on the right-hand side of this equation, we see that we’re trying to add a distance with units of meters to a time with units of seconds. And since we’re trying to add together two quantities with different units to each other, this doesn’t meet our second requirement on the units of a formula. So we know that option (C) can’t be correct.

If we now consider option (D), we see that this formula is telling us that speed 𝑠 is equal to distance 𝑑 multiplied by time 𝑑. If we look at this formula in terms of its units, we see that on the left-hand side we have speed with units of meters per second. Meanwhile, on the right-hand side, we have distance with units of meters multiplied by time with units of seconds. So on the left-hand side, we have units of meters divided by seconds, while on the right-hand side, we have meters multiplied by seconds. So the units on the two sides of the formula don’t agree with each other. And our first requirement on the units of a formula is not met by the formula in option (D). And so we know that this cannot be the correct answer.

This leaves us with one last formula to consider. And that’s option (E). This formula states that speed 𝑠 is equal to distance 𝑑 divided by time 𝑑. Now at this point, we could say that this must be the correct answer by process of elimination since we’ve already shown that all of the other answers cannot be correct. But in order to be thorough, we should check that the units do make sense in this case.

If we look at the left-hand side of this formula, we see that we have speed with units of meters per second. And on the right-hand side, we have distance with units of meters divided by time with units of seconds. On the left-hand side of this, we have units of meters per second. Now units of meters per second is nothing more than the fraction units of meters divided by units of seconds. And so the units on the left- and the right-hand side of this formula do agree with each other. And so this formula meets our requirements that we have for the units.

And so we have our answer that the formula that correctly relates time, distance, and speed is given by option (E), 𝑠 equals 𝑑 divided by 𝑑.

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