Hello, this lesson is about measuring the speed of water waves from measurements of distance and time.

It's from the physics unit Measuring Waves.

My name is Mr. Fairhurst.

By the end of this lesson you should be able to explain how to use measurements of distance and time to calculate the speed of water waves accurately.

And in doing so, you should become much more familiar with these terms and these are the keywords that you're going to come across during the lesson.

If at any point in the lesson you want to come back and look at the definitions of those key terms, pause the video and come back to this slide and have a look.

Now this lesson is split into two parts.

The first part we're going to look at how you can measure the speed of water waves, and this part of the lesson will finish with a practical activity in which you're going to measure the speed of water waves.

And then in the second part of the lesson we're going to see how you can use repeat measurements to improve the accuracy of your final measurements.

Okay then, so let's make a start.

What we know about a water wave is that the speed of that wave will vary depending on the depth of the water.

That's because if the water changes the depth, the water wave is travelling through a different medium and therefore its speed is going to be affected.

Now we can measure the speed at different depths accurately and easily if we use laboratory conditions and set up the measurements so we can make them easy and straightforward.

Now to do that, we can use water in a plastic tray with straight edges, and if we lift one end and drop it onto the table, then that will cause a water wave with a straight edge to go forward and backwards along the length of the tray, which is easily measured.

Okay, so have a quick go at this true and false question.

Pause the video while you do so and start again once you're ready.

Okay, so how did you get on? It is true that laboratory conditions make it easy to measure accurately the speed of a water wave at a certain depth.

And the reason is reason A, if you go into nature, into a river or the sea, the depth of the water often varies.

And in the laboratory we can make sure that we keep the depth the same all of the time so that we can measure the speed in a particular depth much more accurately.

And we measure the speed using this equation, it's the speed equation.

Speed equals distance divided by time.

And we need to take measurements of distance and time off our wave in the tray in order to calculate the speed.

We can measure the length using a metre ruler and we can measure that up to the nearest millimetre or to the nearest 0.

1 centimetre because that's the smallest division on the ruler.

And when we write that down, we write 35.

3 centimetres, which is our measurement here, plus or minus 0.

1 centimetres.

That's indicating how accurately we can take the measurement to.

It's the nearest small division on the ruler.

To measure the time that a wave takes to travel the length of a tray, we use a timer.

Now if we're timing with a stopwatch, we can start it or stop it a moment too late or a moment too soon, and that can introduce errors and uncertainties in our measurement.

And the difference between the actual time that the wave takes to travel and the measured time, which includes any mistakes that we've made are caused by what we call random errors.

Sometimes we can start the stopwatch too soon, sometimes we can start it a little bit too late, so a random error can be too big or too small.

Okay, just have a quick go at this question.

Which of these is best described as a random error? Pause video for a moment and start again once you've made your selection.

Okay then which of these best described a random error? And the answer is A, it's a person, it's made by a person measuring the temperature wrongly.

They might misread the scale a little bit too high or a little bit too low.

Now when we take a measurement, it's pretty much impossible to know for sure exactly the size of the error in that measurement.

But what we can do is we can take a set of repeat measurements like these ones for time and use those to estimate the uncertainty in the measurement caused by random errors.

Now for these measurements of time, the mean average is 0.

50 seconds.

The smallest reading is 0.

45 and the largest is note 0.

55.

So we can guess that the smaller reading is a little bit too small, the larger reading is a little bit too big, and we can work out the difference between the largest and the smallest measurement and call that the range of measurements.

In this case it's 0.

10 seconds.

And what we can do with those measurements, we can use those to estimate the size of the uncertainty and we say it's equal to half of the range, or 0.

05 seconds.

Now in this particular example the larger measurement is 0.

05 seconds higher than the mean average and the smaller measurement is 0.

05 seconds below the mean average.

So that's the variation in our results compared to the mean.

So we can use that.

It's not saying that all the errors are exactly 0.

05 seconds, but from our repeat measurements that is the likely size of the maximum uncertainty in our measurements.

We write that down as the time measured is 0.

50 seconds plus or minus 0.

05, which is our uncertainty.

Have a look at this question.

Look at the results and decide from the options which one of those is the size of the uncertainty in these measurements shown.

Pause the video and start again once you've got your answer.

Okay, so how did you get on? The uncertainty is equal to half of the range.

The range is the biggest reading, 1.

20, take away the smallest, 1.

08 seconds, which gives us a range of 0.

12 seconds, and the uncertainty is estimated to be equal to half of that range, or 0.

06 seconds.

So well done if you've got that answer.

If we're getting random errors in our measurements, one thing that we need to think about is how can we make those errors a little bit smaller? How can we reduce the size of those so we get more accurate results? Now with our experiment at the moment, one way we can do that is to film the wave moving forwards and backwards with a timer.

If we're timing the time the wave takes to get to either end of the tray, then what we can do is we can play that film back in slow motion, maybe use freeze frame when it reaches the end of the tray, and record the exact time shown on the timer in order to take our measurements.

And that's going to mean that we don't rely quite so much on our reaction times.

Another way to reduce the size of the error is to not just measure the wave going one length of the tray, but to measure it going two or three lengths of the tray.

And if we do that, we will need to divide our final measurement of time by three in order to get the time for one length of the tray.

But when we divide the time by three, we also divide any errors by three, any mistakes by three, so we get three times more accurate results by measuring three times longer distance.

Have a go at this question.

What does not reduce the random error when measuring the time it takes for one full swing backwards and forwards and back to where you started from? Pause the video whilst you have a go at this question and start again once you're ready.

Okay, so what do you think? Which does not reduce the random error? And the correct answer was repeating the measurement.

We can estimate the size of the error if we take repeat measurements, but we can't reduce it.

What we can do is we can film the swing with a timer in view.

So we take away that reaction time from the equation.

And we can also time five swings and divide by five.

So that would divide the size of our error by five as well.

So both of those would be good ways to reduce random error.

Well done if you've got this question correct.

What I'd like you to do now is to have a go at that investigation to measure and compare the speed of a waterway for three different depths in a plastic tray and to use some of those strategies that we've talked about to reduce the size of the errors involved.

For each depth, I'd like you to take three repeat measurements and to measure at least three different depths, probably between one and two centimetres deep.

If you have more time, by all means take a few extra depths of readings.

Pause your video whilst you do that and start again once you've got all of your results and are ready to move on.

Okay, so hopefully you've got a good set of results from your investigation.

Here's a sample set of results that I took that we're going to use later on in the lesson.

They won't be exactly the same as your results, but yours should be fairly similar.

Right, we're going to move on to part two of the lesson now using repeat measurements to improve the accuracy.

So we've taken repeat readings for our investigation, let's see how we can use those to improve accuracy in our final results.

When we take repeat measurements, sometimes we get clear mistakes that are very different from the other measurements.

And these we call anomalous results.

Here's a set of readings that include an anomalous result.

Three of the readings are fairly similar and one is significantly different and that reading, 0.

92 seconds, is anomalous.

It's a clear mistake, so we get rid of that answer.

We cross it out by putting a line through, making sure we can still read the numbers in case we need to come back later.

Occasionally in science we find that an anomalous result actually is quite an important one.

So always record your anomalous results, check them if you can.

So in this case, rather than stop at three results, I noticed this anomalous result as it was taking the the third result.

So I took a fourth repeat reading just to double check that it really was an anomalous result.

Have a look at these results yourself and what I'd like you to do is to spot which of these measurements was an anomalous result.

Pause the video whilst you do so and start again once you're ready.

Okay, so which one did you see? It was 3.

86 seconds here.

It was not too different from the other two, but it was significantly different.

It was a good 0.

3 or 4 seconds less than the other two readings.

So well done if you've got that one right.

This one also was wrong.

It's a little bit harder to see.

They're all three point something, but 3.

04 is again, it's nearly 0.

2 of a second less than the other two readings, so we can cross that one out.

And in this instance it might be sensible to take a fourth repeat reading.

So well done if you spotted that one as well.

And with the final line of results, they're all relatively close together and I don't think any of those is an anomalous result.

Now as I've just said, anomalous results are not used to calculate the mean.

We cross them out and we don't use them because they're clearly mistakes.

Here's a set of readings that we had before with the anomalous results crossed out, and what we need to do now is to calculate the mean value.

We know that each of the values we've measured might be a little bit too high or a little bit too low, so we get a more accurate value if we calculate the mean.

Now for the eight millimetre depth, we calculate the mean by adding the values together and dividing by the number of values that we've added together.

So in this case, we've added them together and divided by two, and our mean value is 4.

175 seconds.

We write that down as 4.

18 seconds, which is as accurate as we've taken the measurements to.

We can't give a mean average that's more accurate than our actual measurements.

If we do the same for the next two rows, we get a mean of 3.

19 seconds, which rounds up from the average of 3.

185, and for the final row it was 2.

35 seconds.

So there are mean average values for each of those depths.

If you use your results, you'll get slightly different values, but that's okay.

Have a look at this question.

Laura takes some measurements of time for another water wave.

What is the mean of her times? Be careful about any anomalous results.

Pause the video, calculate your answer, and start again once you've got it.

Okay, so how did you get on? First of all, we need to spot any anomalous results, and in this instance we've got one there.

0.

92 is clearly wrong.

So we cross that out and we do not use it to calculate the mean value.

We add those three other values together, divide by three, and the mean average of those values is 0.

49 seconds.

So well done if you correctly got rid of the anomalous result and calculated the mean correctly.

Now the mean of several measurements is likely to be closer to the true value than any single measurement.

Here's a set of measurements to measure Jacob's height.

Some of them are a little bit too high, some of them are a little bit too low, but on average the mean average is going to be much closer to Jacob's height.

You might by chance get one of those measurements that is exactly his height, but you don't know which one.

So taking an average of several measurements is a better way to be more certain that you are closer to his actual height.

So that's the mean height of these values.

And as you can see, it is close to his actual height, but not quite exactly his actual height.

Have a look at this question.

Pause a video whilst you answer the question and start again once you're ready.

Okay, so how did you get on? Which of these are reasons why repeat measurements usually give more accurate results? Well, B is a correct answer.

Any single measurement might be very different to the true value.

We don't know when we take a single measurement, if it's exactly correct or whether it's very different from what we're actually trying to measure.

We need to take more measurements to check.

And if we take more measurements and we have made a big mistake, we can spot any anomalous results, cross those out, and not use them in calculating our mean average.

We can spot any completely silly mistakes that we've made easily if we take repeat measurements.

And D is also correct.

All of our measurements can be a little bit too big or a little bit too small, and we're never sure which.

They might be spot on, but again, we're never sure which so that is another reason why taking more readings is better, because we can then see which readings are too small and which are too big.

What I'd now like you to do is to look at these results and for each depth to calculate the mean value of the results, to calculate the uncertainty in each of those mean times that you've calculated, and also to calculate the speed of a water wave for each depth measuring to centimetres per second.

Just pause the video whilst you do that and once you've got all of your answers, start it again and we can check those results.

Okay, so how did you get on? Let's start with the 10 millimetre depth.

Looking at those results, they're all very close together, so we simply add the three results together and divide by three.

And the mean value we get is 3.

67.

We've rounded it to the same number of significant figures as each of the measurements, because that's how accurately we can calculate it to.

For 16 millimetre depth there is an anomalous result, and that one is 2.

88 seconds.

The other two are very, very close together and the 2.

88 is quite different.

Just be a little bit careful here because it's tempting to say that the 3.

02 seconds is the anomalous result because it's just flipped over into the next second.

The mean value of those two results is 3.

00 seconds.

Again, putting.

00 to show that the accuracy is just the same as for the measurements that we took to get to calculate that mean value.

And then on the third row, for 22 millimetre depth, we add those three results together, divide by three, and round to three significant figures again.

So 2.

49 is the mean there.

To calculate the uncertainty, we need to first of all work out the range.

So for 10 millimetres the range is between 3.

65 seconds and 3.

70 seconds, so the range is 0.

05 seconds.

Half of the range is equal to the uncertainty, and half that range is 0.

025.

Now that is a little bit more accurate than what we've got there, so we need to round up to 0.

03 So we said the uncertainty is naught point, plus or minus 0.

03 seconds.

We can do the same for the other two rows.

We get plus or minus 0.

02 seconds and we get plus or minus 0.

03 seconds.

Now to calculate the speed, we could either divide the time by three and use the length of the tray we've got there.

But what I'll do instead, I'll multiply the length of the tray by three, so we get the total distance for those times.

And speed is distance divided by time, and the answers we get are 31.

3 centimetres per second, 38.

3 centimetres per second, and 45.

8 centimetres per second.

And again, you'll notice I've rounded all of those answers to three significant figures, which is equal to the same accuracy that we took our measurements to.

So very well done if you got all of those calculations correct.

Well done to make it to the end of the lesson.

This is a short summary slide that covers the main points that we've gone through during the lesson.

We use the equation speed equals distance divided by time to calculate the speed of water waves by measuring first the distance and then the time taken for the waves to travel that distance.

We took repeat readings so we could check for mistakes in taking each measurement.

And any mistakes that we noticed, any results that were quite different from these results were the anomalous results.

Measurements still vary even if no mistakes are made and the differences are sometimes caused by random errors that can lead to an uncertainty in a measurement.

And the uncertainty we estimated as being equal to half of the range of the repeated measurements.

And the mean value, the mean average, of the repeated measurements usually gives a much more accurate measurement than any single measurement that we can take.

So well done again for reaching the end of the lesson, I do hope to see you next time.

Goodbye.