Lesson video

Hi, I'm Mrs. Wheelhouse, and welcome to today's lesson that's from the unit "Maths in the Workplace." In this series of lessons, we're exploring how maths is used in different careers.

So, let's get started.

In today's lesson, we're gonna be considering some of the maths that are used by people who work as astrophysicists.

Let's get started.

Now, an astrophysicist tries to understand the universe and its contents by applying the laws of physics.

Is that something you already knew? I mean, well done if you did, but it's not necessarily a career that everyone knows about.

So let's explore how maths is used by astrophysicists.

We're gonna begin by looking at what being an astrophysicist involves.

An astrophysicist tries to understand the universe and its contents by applying the laws of physics.

What do you mean by "the universe and its contents?" Or as Aisha says, "What are the contents of the universe?" Well, let's consider what that could be.

It could be galaxies, stars, planets, moons, comets, asteroids.

The list goes on.

These are the contents of the universe, and it's not even all of it.

So let's do a quick check.

An astrophysicist tries to understand the universe and its contents by applying: A, the laws of biology, B, the laws of chemistry, C, the laws of economics, or D, the laws of physics.

Pause the video and make your choice now.

Welcome back.

You should, of course, have chosen D, the laws of physics.

Oh, that's an interesting question, Aisha.

Are astrophysicists and astronomers the same thing? So Aisha thinks, well, are they similar because they start with the word astro, so maybe they are the same thing? Hmm.

Izzy says, "Well, yes and no." I wonder what she means by that.

Astrophysics is a branch of astronomy.

Astronomers study the universe using observational techniques to assess and categorise various celestial bodies.

Astrophysicists have a more narrowed focus.

They study the physics of the universe, such as how the celestial bodies interact or other physical processes.

Astrophysicists often have to interpret data.

This may involve converting between different types of unit of measurement and using formulas to find new data.

Standard form may be used if the values recorded are extremely large.

Let's do a quick check to remember what standard form is.

Standard form is when a number is written in the form: Now you have four options.

Read each option carefully and then choose the one where you could see standard form displayed correctly.

Pause and do this now.

Welcome back.

You should have picked B.

This one we have A, where A is greater than or equal to one and strictly less than 10.

And N is an integer.

That's the correct way to write a number in standard form.

So let's check you've got that.

Please select the correct way to write this number in standard form.

Pause and do this now.

Which option did you go for? You should have chosen A.

Well done if you did.

It's time now for your first task.

Question one, I'd like you please to fill in the gaps.

Now, the words that are missing can be seen below.

Pay careful attention to the second line of text, you can see there that it reads, blank.

Study the universe using blank, blank.

So there are two spaces there that you'll need to fill in.

Pause the video and have a go of filling this in now.

Welcome back.

Let's look at our second question.

So for question two, part A, the table below shows the approximate average distances from Earth to some of the other planets in our solar system.

What I'd like you to do, please, is write all of those distances in standard form.

Pause the video and do this now.

Welcome back.

Now for question two, part B, in 2021, the fastest speed recorded by a spacecraft was 163 kilometres per second.

Wow, that's fast.

Now using that speed and the formula that distance is equal to speed multiplied by time, please calculate how long it would take, in days, to travel from Earth to each of the following planets.

Round your answer appropriately to the nearest day.

So you might want to think about what that means.

Pause the video and have a go now.

Welcome back.

Let's go through some answers.

So the question one asked you to fill in the gaps.

You should have filled it in as follows: Astrophysics is a branch of astronomy.

Astronomers study the universe using observational techniques to assess and categorise various celestial bodies.

Astrophysicists have a more narrowed focus.

They study the physics of the universe, such as how the celestial bodies interact or other physical processes.

Question two, part A, you needed to fill in the table by writing the distances in standard form.

So reading down the table, you should have 9.

1691 times 10 to the power of 7, 4.

14 times 10 to the power of 7, 7.

834 times 10 to the power of 7, 6.

2873 times 10 to the power of 8, and then lastly, 1.

275 times 10 to the power of 9.

If you need to, please feel free to pause the video so that you can check these carefully against your own work.

And lastly, question two, part B.

So I asked you to use the speed that I gave, and the formula, to calculate the time it would take in days to reach each of the various planets if you started at Earth.

So what I've done here, is shown what it is for Mercury.

Now when I did the calculations for Mercury, I got the value 6.

5 days at the end.

Now, I asked you to round to the nearest day.

It's not gonna take me six days to get to Mercury, it's gonna take more than that, so I need to round up to seven days.

Using the same principles for Venus, Mars, Jupiter, and Saturn, I got the following time in days: So for Venus, it will take three days.

Mars, it will take six days.

Jupiter will take 45 days.

And for Saturn, 91 days.

And that's assuming you're going at the fastest speed recorded by a spacecraft in 2021.

It's now time to look at how astrophysicists determine certain terrain features.

Let's see what we mean by that.

The first Moon landing took place in 1969.

Since humans could walk on the Moon, it meant measurements could be taken on the Moon's actual surface.

What sort of measurements might be useful to record? What did you come up with? Well, let's see if it's the same.

Well, calculating the height of various features, such as mountains, is often done using a laser altimeter.

However, not all space missions have one.

In those cases, we can still determine an estimate for the height of various features using trigonometry.

That's right, using trigonometry.

Simulations can be run on Earth to help study features on other planetary bodies.

This is Professor Veronica Bray.

Now, she's an associate research professor.

She's dressed like this because she's carrying out a simulation with her colleagues.

And you can see them there.

They are simulating the Artemis astronaut conditions at the South Pole of the Moon.

Professor Bray and her colleagues are recreating the correct lighting conditions for astronaut training.

And here they are.

They need to know the sun angle so they can calculate the correct ratio of the height of the light source and the distance from the feature.

Do you recognise that triangle? That's right, it's a right angle triangle.

Mountains on the moon were formed through massive asteroid impacts.

These impacts resulted in the craters we can see, and the walls of these craters can be very high.

To estimate the height of the crater wall at the selected point, we can use trigonometry.

So we'll need to know the length of the shadow and the angle the sun is at, since it's similar to what Professor Bray and her colleagues were working on.

A sketch will simplify what we are considering, so get prepared to see that right angle triangle, just like you saw in the image of Professor Bray and her colleagues.

So, it's only a sketch, remember? So I don't need to be super accurate here.

There's roughly the sun, the surface of the planet, the terrain feature, and there's the sunlight coming down.

Remember, I want to know the length of the shadow and the angle the sun is at.

So there's the length of the shadow, and there would be the angle the sun is at.

Now the length of the shadow can be measured from an image as long as the scale is known, and the angle of the sun can be determined if we know when the image was recorded.

Now given the length of the shadow and the angle of the sun are known, what can we use to determine the height of the feature? So would I use Pythagoras' theorem, similar triangles, or trigonometry? Pause and make your choice now.

Welcome back.

You should, of course, have chosen trigonometry.

I can't use Pythagoras because I would only know one length of my right angle triangle.

And I can't use similar triangles because I don't have the features of another right angle triangle which is similar, so I can't determine multiplicative relationship, so it has to be trigonometry.

Let's do an example, and then you'll do one by yourselves.

When the angle of the sun is at 42 degrees, the length of the shadow of the terrain feature is recorded as 7,300 metres.

What I want to know is what is the height of that crater wall, therefore? So what I've done is I've done a quick sketch, just so I can see where those values go.

I multiply the adjacent, so that's 7,300, by the tangent of the angle, so tan 42, and that gives me the height of the terrain feature, or in this case the height of the crater wall, to be approximately 6.

6 kilometres.

I could, of course, have left it in metres and said it's approximately 6,600 metres, but I've gone for kilometres.

It's your turn now.

When the angle of the sun is 87 degrees, the length of the shadow is recorded as 346 metres.

What is the height of the crater wall? Pause the video and have a go now.

Welcome back.

What did you get? Well, if you did a sketch like I did, you can see where your values went.

And then, again, you had to multiply the adjacent by the tangent of the angle.

So 346 multiplied by tan 87 gives us 6602.

0732 etc, etc, which, again, is approximately 6.

6 kilometres.

Look at that.

The angle of the sun changed and therefore the length of the shadow changed.

But that didn't change the height of the terrain feature, did it? The crater wall was still approximately 6.

6 kilometres high.

It's now time for the second task.

For question one, I'd like you to calculate the value, to one decimal place, of the unknown measurement for each case.

That's for part A, B, C, D, E, and F.

Pay careful attention to which values I've given you and which value I'm asking you to work out.

Pause and do this now.

Welcome back.

Time for question two.

The wall of a crater on Mars is 1.

2 kilometres tall.

When the wall casts a shadow with length 557 metres, what angle, to one decimal place, is the sun at? For question three, another crater on Mars has a wall which casts a shadow of length 21 metres.

This occurs when the sun is at an angle of 67 degrees.

What is the height, to two decimal places, of this wall? Pause the video now while you work these out.

Welcome back.

Time for question four.

The largest crater on the near side of the Moon is called Clavius.

The walls of this crater are 6.

6 kilometres high.

Calculate three pairs of values for the angle of the sun and the corresponding length of the shadow cast by the crater wall.

So I'll give you an example here.

I've said when the angle of the sun is 42 degrees, the length of the shadow is recorded as 7,300 metres.

So with that angle for the sun and that length of shadow, I would be able to calculate that the height of the wall of the crater would be 6.

6 kilometres high.

Another pair of values that work is when the angle of the sun is at 87 degrees and the length of the shadow is recorded as 346 metres.

It's now your turn to come up with three pairs of values that also work.

Pause the video while you do this now.

Welcome back.

The final question now.

In order to calculate an estimate for the height of a crater wall, we've produced a model or sketch of the situation.

Please state an assumption that we've made that may affect the accuracy of our estimated value.

Pause the video and have a go now.

Welcome back.

Let's go through those answers.

So for question one, I asked you to calculate the value, to one decimal place, of the unknown measurement in each case.

For part A, you should have had 19.

3 centimetres.

For B, 28.

2 centimetres.

For C, 162.

6 centimetres.

And for part D, 12.

8 centimetres.

Part E, 18.

7 degrees.

And F, 71.

3 degrees.

Well done if you've got those all right.

Question two, when the wall casts a shadow with length 557 metres, what angle, to one decimal place, would the sun have been at given that the wall of the crater on Mars is 1.

2 kilometres tall? Well, you should have been able to calculate that the angle of the sun would be 65.

1 degree, to one decimal place.

And for question three, I said there's another crater on Mars which has a wall which casts a shadow of length 21 metres when the sun is at an angle of 67 degrees.

What's the height, therefore, to two decimal places, of this wall? Well, I multiply, remember, the adjacent by the tangent of the angle, giving us 49.

47 metres, to two decimal places.

Question four.

Now there are infinitely many pairs that will work here.

You need to check your values are correct, and you can do that by doing the following: Take the length of the shadow that you suggested and multiply it by the tangent of the angle that you suggested.

Your answer should round to 6.

6 if you gave your length of shadow in kilometres, or 6,600 metres if you gave your length of shadow in metres.

Now, you might want to pause here and just check that calculation with your pairs of values to make sure that you do get what you're supposed to get.

So feel free to pause the video now, while you do this.

Now, question five.

In order to calculate an estimate for the height of a crater wall, we've produced a model or sketch of the situation.

I asked you, please, to state an assumption that we've made that may affect the accuracy of our estimated value.

Now, here's just one possible assumption, and you may have others, and it's great if you do.

It doesn't mean that your answer is wrong because it's not the same as mine.

There were lots of possible answers here.

Or one of the possible assumptions could be that we've assumed that the surface of the planet was flat when it's actually curved.

Or, think about how that would make a difference when we've measured it may not be a flat surface.

You might want to have a go at seeing what effect that could have.

So maybe consider taking a ball, placing just a bit of paper or attaching something to that ball so that you can cast a shadow with a light source, and do some measurements to see what effect that curved surface has.

It's time to state what we've learned today.

An astrophysicist tries to understand the universe and its contents by applying the laws of physics.

Astrophysicists often have to interpret data.

This may involve converting between different types of unit of measurement and using formulas to find new data.

Trigonometry can be used to estimate the height of different terrain features on other celestial bodies.

And you saw, didn't you, Professor Bray and her colleagues carrying out a simulation so they could check what measurements were needed when the astronauts go on their mission? Well done today, you've worked really well.

I hope you've enjoyed learning about astrophysicists and how maths is useful to them.

I look forward to seeing you again, for more lessons.

Bye for now.