Video: Calculating Average and Root-Mean-Square Speed

Five bicyclists are riding at the following speeds: 5.40 m/s, 5.70 m/s, 5.80 m/s, 6.00 m/s, and 6.50 m/s. What is their average speed? What is their root mean square (rms) speed?

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Video Transcript

Five bicyclists are riding at the following speeds: 5.40 metres per second, 5.70 metres per second, 5.80 metres per second, 6.00 metres per second, and 6.50 metres per second. What is their average speed? What is their root mean square, RMS, speed?

Okay, so in this question, we’ve been given five different speeds. And we need to find their average speed and the root mean square, or RMS, speed. Let’s start by finding their average speed.

We can call this 𝑉 bar. And the way to find the average of multiple values is to add up all of the values and divide by how many there are. So in this case, we’ve got a value of 5.40, 5.70, 5.80, 6.00, and 6.50 metres per second. We have to add all of those up and then divide by however many there are, which in this case is five. And so that’s exactly what we’ve done here.

We’ve also factorised out the metres per second unit on the right-hand side, so that we don’t have to faff about with the units whilst we’re doing the calculation. We also know that the final answer is going to be in metres per second because we’re calculating an average velocity.

So plugging these numbers into our calculator gives us a value of 5.88 metres per second. And this number is fine to two decimal places because all of the values we’ve been given in the question have been given to two decimal places. And that is the answer to the first part of our question.

Let’s move on to finding the root mean square speed. We can call this 𝑉 sub RMS for root mean square. Now the root mean square speed is the root or the square root of the mean of the square speeds. In other words, what we need to do is to first square each one of these quantities, then find the mean value of those squared speeds, and then find the square root of that value. And that looks something like this, the square root of the mean of the squared speeds.

Now the mean of the squared speeds is simply found by adding together all of the squared values, as we’ve done here, and then dividing by how many there are, which is five once again. Now everything inside this set of parentheses is the mean square speed. So to find the root mean square speed, we simply need to take its square root.

Now we can plug this whole thing into our calculator to give us a root mean square speed of 5.89 metres per second. And once again, this is fine to two decimal places because all of the values we’ve used are to two decimal places. And so that’s the answer to the second part of our question.

Before we go, notice the difference between the average speed and the root mean square speed. The average speed is 5.88 metres per second, whereas the root mean square speed is 5.89 metres per second. So these two quantities, the average speed and the root mean square speed, are not the same.

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