Maths, but not as you know it… (St Edmund Hall Oxford Magazine)

Dr Tom Crawford joined the Hall in October 2018 as a Stipendiary Lecturer in Mathematics, but he is far from your usual mathematician…

Tom’s research investigates where river water goes when it enters the ocean. A simple question, you might first think, but the complexity of the interaction between the lighter freshwater and the heavier saltwater, mixed together by the tides and wind, and pushed ‘right’ along the coast due to the Earth’s rotation, is anything but. The motivation for understanding this process comes from recent attempts to clean-up our oceans. Rivers are the main source of pollution in the ocean, and therefore by understanding where freshwater ends up in the ocean, we can identify the area’s most susceptible to pollution and mitigate for its effects accordingly.

To better understand this process, Tom conducts experiments in the lab and has conducted fieldwork expeditions to places as far-flung as Antarctica. What the southern-most continent lacks in rivers, it makes up for in meltwater from its plethora of ice sheets. The ultimate process is the same – lighter freshwater being discharged into a heavier saltwater ocean – and as the most remote location on Earth the influence of humans is at its least.

If you thought that a mathematician performing experiments and taking part in fieldwork expeditions was unusual, then you haven’t seen anything yet. Tom is also very active in outreach and public engagement as the author of the award-winning website tomrocksmaths.com which looks to entertain, excite and educate about all thing’s maths. The key approach to Tom’s work is to make entertaining content that people want to engage with, without necessarily having an active interest in maths. Questions such as ‘how many ping-pong balls would it take to raise the Titanic from the ocean floor?’ and ‘what is the blast radius of an atomic bomb?’ peak your attention and curiosity meaning you have no choice but to click to find out the answer!

Tom is also the creator of the ‘Funbers’ series which was broadcast on BBC Radio throughout 2018 telling you the ‘fun facts you didn’t realise you’ve secretly always wanted to know’ about a different number every week. From the beauty of the ‘Golden Ratio’ to the world’s unluckiest number (is it really 13?) via the murderous tale of ‘Pythagoras’ Constant’, Funbers is a source of endless entertainment for all ages and mathematical abilities alike.

And now for the big finale. If you are familiar with Tom’s work, you may know where we are heading with this, but if not, strap yourself in for the big reveal. Dr Tom Crawford is the man behind the ‘Naked Mathematician’ (yes you did read that correctly). To try to show that maths isn’t as serious as many people believe, to try to engage a new audience with the subject, and just to have fun, Tom regularly gives maths talks in his underwear! His ‘Equations Stripped’ series on YouTube has reached 250,000 views – that’s a quarter of a million people that have engaged with maths that may otherwise have never done so. His recent tour of UK universities saw several thousand students come to a maths lecture of their own accord to learn about fluid dynamics. It may not be to everyone’s tastes, but our current methods of trying to engage people with maths are failing, so why not try something new? This is maths, but not as you know it.

You can find all of Tom’s work on his award-winning website and you can follow him on FacebookTwitterYouTube and Instagram @tomrocksmaths for the latest updates.

The original article published in the Aularian magazine can be found here.

What is the blast radius of an atomic bomb?

Picture the scene: you’re a scientist working for the US military in the early 1940’s and you’ve just been tasked with calculating the blast radius of this incredibly powerful new weapon called an ‘atomic bomb’. Apparently, the plan is to use it to attack the enemies of the United States, but you want to make sure that when it goes off any friendly soldiers are a safe distance away. How do you work out the size of the fireball?

One solution might be to do a series of experiments. Set off several bombs of different sizes, weights, strengths and measure the size of the blast to see how each property affects the distance the fireball travels. This is exactly what the US military did (see images below for examples of the data collected).

Picture1 Picture2

These experiments led the scientists to conclude that were three major variables that have an effect on the radius of the explosion. Number 1 – time. The longer the time after the explosion, the further the fireball will have travelled. Number 2 – energy. Perhaps as expected, increasing the energy of the explosion leads to an increased fireball radius. The third and final variable was a little less obvious – air density. For a higher air density the resultant fireball is smaller. If you think of density as how ‘thick’ the air feels, then a higher air density will slow down the fireball faster and therefore cause it to stop at a shorter distance.

Now, the exact relationship between these three variables, time t, energy E, density p, and the radius r of the fireball, was a closely guarded military secret. To be able to accurately predict how a 5% increase in the energy of a bomb will affect the radius of the explosion you need a lot of data. Which ultimately means carrying out a lot of experiments. That is, unless you happen to be a British mathematician named G. I. Taylor…

Taylor worked in the field of fluid mechanics – the study of the motion of liquids, gases and some solids such as ice, which behave like a fluid. On hearing of the destructive and dangerous experiments being conducted in the US, Taylor set out to solve the problem instead using maths. His ingenious approach was to use the method of scaling analysis. For the three variables identified as having an important effect on the blast radius, we have the following units:

Time = [T],       Energy = [M L2 T-2],      Density = [M L-3],

where T represents time in seconds, M represents mass in kilograms and L represents distance in metres. The quantity that we want to work out – the radius of the explosion – also has units of length L in metres. Taylor’s idea was to simply multiply the units of the three variables together in such a way that he obtained an answer with units of length L. Since there is only one way to do this using the three given variables, the answer must tell you exactly how the fireball radius depends on these parameters! It may sound like magic, but let’s give it a go and see how we get on.

To eliminate M, we must divide energy by density (this is the only way to do this):

eqn1.png

Now to eliminate T we must multiply by time squared (again this is the only option without changing the two variables we have already used):

eqn2.png

And finally, taking the whole equation to the power of 1/5 we get an answer with units equal to length L:

eqn3

This gives the final result that can be used to calculate the radius r of the fireball created by an exploding atomic bomb:

eqn4

And that’s it! At the time this equation was deemed top secret by the US military and the fact that Taylor was able to work it out by simply considering the units caused great embarrassment for our friends across the pond.

I love this story because it demonstrates the immense power of the technique of scaling analysis in mathematical modelling and in science in general. Units can often be seen as an afterthought or as a secondary part of a problem but as we’ve seen here they actually contain a lot of very important information that can be used to deduce the solution to an equation without the need to conduct any experiments or perform any in-depth calculations. This is a particularly important skill in higher level study of maths and science at university, as for many problems the equations will be too difficult for you to solve explicitly and you have to rely on techniques such as this to be able to gain some insight into the solution.

If you’re yet to be convinced just how amazing scaling analysis is, check out an article here explaining the use of scaling analysis in my PhD thesis on river outflows into the ocean.

And if that doesn’t do it, then I wish you the best of luck with those atomic bomb experiments…

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