100 Essential Things You Didn't Know You Didn't Know (17 page)

Clothoids have another nice feature that has led to their incorporation into the design of complex motorway junction exits or railway lines. If a car is being driven along a curving motorway exit road, then as long as the driver keeps to a constant speed you can simply move the steering wheel with a constant rotation rate. If the bend were a different shape, then you would need to keep adjusting the rate of movement of the steering wheel or the speed of the car.

51

A Taylor-made Explosion

I do not know with what weapons World War III will be fought, but World War IV will be fought with sticks and stones.

Albert Einstein

The first atomic bomb exploded at the Trinity test in New Mexico, USA,210 miles south of Los Alamos, on 16 July 1945. It was a watershed in human history. The creation of this device gave human beings the ability to destroy all human life and produce deadly long-term consequences. Subsequently, an arms race saw an escalation in the energetic yield of these bombs as the United States and the Soviet Union sought to demonstrate their ability to produce increasingly devastating explosions. Although only two of these devices were ever used in conflict,
^fn1
the ecological and medical consequences of this era of tests in the atmosphere, on the ground, below ground and underwater, are still with us.

The explosions were much photographed at the time and produced a characteristic fireball and a canopy of debris that came
to
symbolise the consequences of nuclear war. The familiar mushroom cloud
^fn2
forms for a reason. A huge volume of very hot gas with a low density is created at high pressure near ground level – atomic and nuclear bombs were generally detonated above ground to maximise the effects of the blast wave in all directions. Just like the bubbles rising in boiling water, the gas accelerates up into the denser air above, creating turbulent eddies curving downwards at their edges while additional debris and smoke streams up the centre in a rising column. The material at the core of the detonation is vaporised and heated to tens of millions of degrees, producing copious x-rays, which collide with and energise the atoms and molecules of the air above, creating a flash of white light whose duration depends on the magnitude of the initial explosion. As the front of the column rises, it spins like a tornado and draws in material from the ground to form the ‘stem’ of the growing mushroom shape; its density falls as it spreads, and eventually it finds itself with the same density as the air above it. At that moment, it stops rising and disperses sideways, so that all the material drawn up from the ground bounces backwards and descends to create a wide expanse of radioactive fall-out.

Ordinary explosions of TNT, or other non-nuclear weapons, have a rather different appearance because of the lower temperatures created at the start of a purely chemical explosion. This results in a turbulent mix of exploding gases rather than the organised stem and umbrella-like mushroom cloud.

One of the pioneers of the studies of the shape and character of large explosions was the remarkable Cambridge mathematician Geoffrey (G.I.) Taylor. Taylor wrote the classified report on the expected character of an atomic bomb explosion in June 1941. He became well known to a wider public after
Life
magazine in the
USA
published a sequence of time-lapsed photographs of the 1945 Trinity test in New Mexico. The energy yield from this and other American atomic-bomb explosions was still top secret, but Taylor showed how a few lines of algebra enabled him (and hence anyone else with a smattering of simple mathematics) to work out the approximate energy of an explosion just by looking at the photographs.

Taylor was able to work out the expected distance to the edge of the explosion at any time after its detonation by noting that it can depend significantly on only two things: the energy of the explosion and the density of the surrounding air that it is ploughing through. There is only one way that such a dependence can look,
¹³
so approximately:

The published photos showed the explosion at different times since detonation and those times were printed down the side of each photo with a distance scale running along the bottom of the photo to gauge the size. From the first frame of the photo Taylor noted that after 0.006 seconds the blast wave of the explosion had a radius of roughly 80 metres. We know that the density of air is

1.2 kilograms per cubic metre, and so the equation then tells us that the energy released was about 10
¹⁴
Joules, which is equivalent to about 25,000 tons of TNT. For comparison, we know that the 2004 Indian earthquake released the energy equivalent to 475 million tons of TNT.

^fn1
The ‘Little Boy’ bomb dropped on Hiroshima was a fission device with 60kg of uranium-235 and created a blast equivalent to 13 kilotons of TNT, killing approximately 80,000 people immediately; the ‘Fat Man’ bomb dropped on Nagasaki was a fission bomb with 6.4kg of plutonium-239, equivalent to the blast from 21 kilotons of TNT. About 70,000 people were killed instantly.

^fn2
The name ‘mushroom cloud’ became commonplace in the early 1950s, but the comparison between bomb debris patterns and ‘mushrooms’ dates back at least to newspaper headlines in 1937.

52

Walk Please, Don’t Run!

You can spot the northern Europeans because they walk faster than the leisurely art of the paseo strictly requires.

Benidorm travel guide

Walk along a busy city high street and most of the people around you will be walking at about the same speed. A few people are in a bit of a hurry and a few others move very slowly, perhaps because of age, infirmity or totally impractical footwear. When you walk, you keep a part of one foot in contact with the ground all the time and you straighten your leg as you push off from the ground. Indeed, the rules of race walking make these the defining features of walking, which distinguish it from running: a failure to adhere to them results in warnings and ultimately disqualification from a race. As you walk, your hips will rise and fall as your centre moves in a gentle circular arc each time you make a complete stride. So if the length of your leg from the ground to your hips is L, you will be creating an acceleration equal to v
²
/L upwards towards the centre of that circular arc of movement. This cannot become greater than the acceleration due to gravity, g, that pushes us down to the ground (or we would take off!), and so g > v
²
/L and we deduce that, roughly, the top speed for normal walking is about √(gL). Since g = 10 ms
^-2
and a typical leg length is 0.9 metre, the top speed for ordinary walkers is about 3 metres per second – a fairly good estimate – and the taller you are, the larger L will
be
and the faster you will walk, although because of the square root there really isn’t much difference between the walking speeds of people with the usual range of heights.

Another way to interpret this result is to look at people (or other two-legged creatures) trying to get from one place to another as quickly as possible and to ask at what speed they stop walking and break into a run. The critical speed √(gL) is the fastest rate of progress you can expect to make without breaking contact with the ground (‘lifting’ as the race walkers say). Once you start breaking contact you can go much faster, with a maximum speed of about
where S ~ 0.3 m is the difference in length between your straight leg and bent leg when you push off the ground and n ~ 10 is the number of strides you use to accelerate up to full speed.

Race walkers walk much faster than 3 metres per second. The world record for walking 1,500 metres, set by American Tim Lewis in 1988, is 5 minutes 13.53 seconds, an average speed of 4.78 ms
^-1
. This event is rarely walked, and so it is interesting to look at the highly competitive world record for 20 kilometres, the shorter of the two championship events. This was reduced to 1 hr 17 mins and 16 secs by the Russian walker, Vladimir Kanaykin, on 29 September 2007, an average speed of 4.3 ms
^-1
over more than 12.5 miles! These speeds manage to exceed our estimate of √(gL) because race walkers use a much more efficient style of walking than we do when we stroll down the road. They do not rock their centres up and down and are able to roll their hips in a very flexible fashion to produce a longer stride length and higher frequency of striding. This highly efficient movement, coupled with very high levels of fitness, enables them to sustain impressive speeds over long periods of time. The world record holder over 50 km, more than 31 miles, averages more than 3.8 ms
^-1
and will cover the marathon distance (42.2 km) in 3 hours 6 minutes en route.

53

Mind-reading Tricks

Every positive integer is one of Ramanujan’s personal friends.

John E. Littlewood

Think of a number between 1 and 9. Multiply it by 9 and add the digits of this new number together. Subtract 4 from your answer and you will be left with a single-digit number. Next, convert this number to a letter: if your number is 1 it becomes A, 2 becomes B, 3 becomes C, 4 becomes D, 5 becomes E, 6 becomes F and so on. Now think of a type of animal that begins with your chosen letter and imagine that animal as strongly as you can. Hold it vividly in the forefront of your mind. If you look at the note
14
at the back of the book, you will see that I have read your mind and discovered the animal that you were imagining.

This is a very simple trick, and you ought to be able to work out how I was able to guess the animal of your choice with such a high likelihood of success. There is a little mathematics involved, in that some simple properties of numbers are exploited, but there is also a psychological – and even zoological – ingredient as well.

There is another trick of this general sort that involves only the properties of numbers. It uses the number 1089, which you may well already have listed among your favourites. It was the year in which there was an earthquake in England; it is also a perfect square (33×33); but its most striking property is the following.

Pick any three-digit number in which the digits are all different
(like 153). Make a second number by reversing the order of the three digits (so 351). Now take the smaller of the two numbers away from the larger (so 351 − 153 = 198; if your number has only two digits, like 23, then put a 0 in front, so 023). Now add this to the number you get by writing it backwards (so 198 + 891 = 1089). Whatever number you chose at the outset, you will end up with 1089 after this sequence of operations!
¹⁵

54

The Planet of the Deceivers

You can fool some of the people some of the time, and some of the people all the time, but you cannot fool all the people all of the time.

Abraham Lincoln

One of the human intuitions that has been honed by countless generations of social interaction is trust. It is founded upon an ability to assess how likely it is that someone is telling the truth. One of the sharp distinctions between different environments is whether we assume people are honest until we have reason to think otherwise, or whether we assume them to be dishonest until we have reason to think otherwise. One encounters this distinction in the bureaucracies of different countries. In Britain officialdom is based upon the premise that people are assumed to be honest, but I have noticed that in some other countries the opposite is the default assumption and rules and regulations are created under a presumption of dishonesty. When you make an insurance claim, you will discover which option your company takes in its dealings with its customers.