Video: Measurement Uncertainties

A distance of 115 m is measured to the nearest meter. The distance is run in a time of 12 s, measured to the nearest second. Rounding to an appropriate number of significant figures, what was the average running speed?

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Video Transcript

A distance of 115 meters is measured to the nearest meter. The distance is run in a time of 12 seconds, measured to the nearest second. Rounding to an appropriate number of significant figures, what was the average running speed?

Alright, so in this example, we have a runner running at distance, which is measured to the nearest meter, to be 115 meters. Along with this distance, the time it takes the runner to cover that distance is also measured. And that time — we can call it 𝑡 — is measured to the nearest second to be 12 seconds. The question is what is the runner’s average speed. And importantly, we want to report this average speed to the appropriate number of significant figures. Along these lines, we can begin by counting the number of significant figures in each of these two numbers, the time 𝑡 and the distance 𝑑. The measure time has one, two significant figures and the measured distance has one, two, three of them.

At this point, let’s recall that the average speed an object has — we’ll call it 𝑠 — is equal to the distance the object travels divided by the time it takes to travel that distance. We can see then that our runners average speed is going to equal the distance, 115 meters, divided by the time, 12 seconds. But the question is, how should we report our answer? After all, our distance has three significant figures, but the time just has two. Should our final answer be in terms of three or two significant figures? To answer this, we can recall a rule for combining numbers that have different numbers of significant figures in them.

That rule goes like this: when combining values with different numbers of significant figures, answer using the smallest number of significant figures. Applying this rule to our situation, we see that our two values have different numbers of significant figures: three and two, respectively. This rule says that as we combine them, like we are in calculating average running speed, that we should keep the least number or the smallest number of significant figures any one of them possesses. That smallest number is two. And this means we’ll round our final answer to be given in terms of two significant figures.

Now, when we calculate this fraction 115 meters divided by 12 seconds, we get this result of 9.583 repeating meters per second. We know this isn’t our final answer because we’ll round it. And in particular, we’ll round it to two significant figures. To do that, let’s count off two significant figures. Starting at the front of the value, nine and five are the first two. And then we look at the next digit in our number, eight. Since eight is greater than or equal to five, that means we’ll go to our previous number, the five, and we’ll round that up one. The five rounds up to a six. And our final answer to two significant figures is 9.6 meters per second. That’s the average running speed reported to two significant figures.

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