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March 2015 Archives

I can't help thinking that the West Indies team got their run chase strategy wrong on Sunday night. They had a tricky task ahead of them. One might say the problem was one of their own making, judging from the rubbish that they served up to Guptill to hit at the end of tne New Zealand innings, but that was in the past. The fact was they had to chase down 393 in 50 overs. How does a team go about doing that?

I would have thought that the obvious answer was 'in the same way that the opposition made it'. In other words, keep the scoreboard turning over nicely in the early stages, but without taking excessive risks, and then, with the wickets in hand (especially Gayle's), hit the accelerator at the end. Rather than taking 50 overs to reach the target, they looked like they were trying to do it in 40. It was never going to work. You can contrast this to the calm manner in which Sri Lanka reached England's 300+ score in the group stages. Nothing flash, no excessive risks, they just ticked the board over just short of the required run-rate, and then with wickets in hand they pushed on at the end to win easily. No fuss, no stupid shots. They knew exactly what was needed, and they achieved it. 

A school of thought says, other things being equal,  it's much better to bat second in a limited overs match, because you know exactly what you have to do. It clarifies your batting strategy: Choose whatever strategy maximizes your chances of reaching the target score within 50 overs. Yes, there are other considerations, our friends Duckworth and Lewis being one, but let's put those aside for the moment.  You have 10 wickets and 50 overs, and a target score. You only need to beat that target by 1 run on the last ball, with one wicket left.  What strategy will maximize your chance of success?

The team batting first has a much less defined problem. They don't know what a winning score is. True, they'll have some feel of what one will be, given the ground, the quality of opposition, the state of the pitch and so on, but fundamentally, they don't know what score is going to be good enough. For example, a team could probably secure a moderate score (let's say 280) with 80% probability, by batting conservatively. Or they could aim for a larger score (say 350) by taking more risks. They might get there with a probability of 25%,  but then there's the possibility (maybe also 25%) they crash out on the way and end up with something more like 250. What  should they do? Which strategy is better? 

It's viewed as a criminal offence if the team batting first fails to bat out its 50 overs. If it's all out beforehand, it has obviously taken too many risks. But also, one could say its a criminal offence if they end up with just a handul of wickets down. In that case they haven't been taking enough risk. Balancing all that up, a typical strategy for a team batting first is to go at a moderate rate to start with, and slowly increase the rate as wickets allow. It seems to have stood the test of time. 

It is possible to make this a bit more mathematical. What a team needs to do is to maximize its score, subject to the constraints that a. it only has 50 overs, and b. it only has 10 wickets. Since the rate of fall of wickets is certainly related to the scoring rate (the higher the runs-per-over, the higher the risk, usually, and the higher the wickets-per-over) we can write down some equations for the situation. [You'll be relieved to hear I won't try to do those here].

It turns out to be is an optimization problem, of which there are many in physics. We can tackle many of them with the Euler-Lagrange equation.  For example, what shape does a chain have when it hangs under its own weight? Take a chain of length 2 metres, secure the ends to posts 1.5 metres apart. The chain clearly sags in the middle, but what shape is the resulting curve?  The chain hangs in such a way as to minimize its gravitational potential energy, subject to the constraint that it has a constant length and that it has to start and end at fixed points. One can put this in equation form and solve for the shape. The resulting curve is called a 'catenary'. 

Certainly, if one could write down the rate of loss of wickets as a function of scoring rate, wickets lost already and other factors, we should be able to have a go at tackling the cricket scoring rate problem. Given the degree of professionalism of the sport, it wouldn't surprise me if that has actually been done by someone. I imagine it would be a closely guarded secret.

 

 

 

 

 

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A mathematician can say what he likes... A physicist has to be at least partly sane

J. Willard Gibbs 

What is it that makes a physicist sane (if only in part)? Everything has to be related back to the 'real world', or the 'real universe'. That is, a physicist has to talk about how things work in the world or universe in which we live, not some hypothetical universe. That's how I think of it, and I know, having done a bit of research with some of my students, a lot of them think the same way. That's not to say mathematicians don't have a lot to say about this universe too. It's just that the constraints on them are somewhat less. 

Another way of looking at it is that physicists work with dimensioned quantities. Most things of physical relevance have dimensions. For example, a book has a length, width and thickness. All of these are distances, and can be measured. The unit doesn't matter; we could use centimetres, inches or light-years - but the physical size of the object is determined by lengths. Also, the book has a mass (one could measure it in kilograms). It might find its way onto my desk at a particular time (measured, for example, in hours, minutes, seconds, millennia or whatever). Perhaps it is falling at a particular velocity - which describes what distance it travels in a particular time. All of these things are physical quantities, and they carry dimensions.

One of my pet hates as a physicist is reading physics material in which the dimensions have been removed. You can do this by writing lengths in terms of a 'standard' length, but then only quoting how many of the standard length it is. So we might talk about lengths in terms of the length of a piece of A4 paper (which happens to be 297 mm); a piece of A2 paper has length 2 standard-lengths, and an area of 4 standard-areas. The problem really comes when the discussion drops the 'standard-length' or 'standard-area' bit and we are left with statements such as a piece of A2 paper has a length of 2 and an area of 4.  It is left to the reader to work out what this actually means in practice. A mathematician can get away with it - she can say what she likes, but not so the physicist. 

Here's a question which illustrates the point? What is the length of a side of a cube whose volume is equal to its surface area? The over-zealous mathematics student blunders straight in there: Let the length be x. Then volume is x^3, and surface area is 6 x^2 (the area of a face is x^2, and there are six on a cube). So x^3 = 6 x^2 ; cancelling x^2 from both sides, we have x=6.  Six what? centimetres, inches, furlongs, parsecs? The point is that the volume of a cube can never be equal to its surface area. Volume and area are fundamentally different things. 

The Wikipedia page on 'fundamental units' , along with many text books, blunders in this way too. The authors should really know better. (Yes, I should fix it, I know...) For example:

A widely used choice is the so-called Planck units, which are defined by setting ħ = c = G = 1

No, NO, NO!  What is wrong with this? How can the speed of light 'c' be EQUAL to Newton's constant of Gravitation 'G". They are fundamentally different things. The speed of light is a speed (distance per unit time), Newton's constant of gravitation is... well.. it's a length-cubed per mass per time-squared. It's certainly not a speed, so it can't possible be equal to the speed of light. And neither can be equal to 1, which is a dimensionless number. What the statement should say, is that c = 1 length-unit per time-unit; and G = 1 length-unit-cubed per mass-unit per time-unit squared. 

However, doing physics can be more complicated that this. A lot of physics is now done by computer. In writing a computer programme to do a physics calculation, we almost always don't have explicit record of the units or dimensions in our calculations. Our variables are just numbers. It's left to us to keep track of what units each of these numbers is in. Strictly speaking, I'd say it's rather slack. It would be nice to have a physics-programming language that actually keeps track of the units as well. However, I'm not aware of one. (If someone could enlighten me otherwise, that would be fascinating...) Otherwise, I'll have to have a go at constructing one.

What's prompted this little piece is that I've been reviewing a paper that has been submitted to a physics journal. The authors have standardized the dimensions out of existence, which makes it awfully hard for me to work out what things mean physically. Just how fast is a speed of 1.5? How many centimetres per second is it? While that might be an answer their computer programme spits out, the authors really should have made the effort of turning it back into something that relates to the real world. In a mathematics journal, they might get away with it. But not in a physics journal. At least, not if I'm a reviewer...

 

 

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Further to my last post, here's a very accessible discussion on some of the physics related to 'the arrow of time'. Maybe, just maybe, Benjamin has the right idea after all...

http://www.bbc.com/earth/story/20150309-why-does-time-only-run-forwards

 

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Benjamin is now two-and-two-thirds, or near enough. As ever, his grasp of physics continues to improve.  In the last few weeks, he has been picking up the idea of time. 

We have a large (more accurately, LARGE) analogue clock on the wall of our lounge. He's watched me take it off the wall, change the battery and move the hands to a new position when it started to run slow. It's clear to Benjamin that what the hands do on the clock is related to the time of day, although just how I think is some way off.  Over the weekend he wanted me to read him a book, but didn't want me to get it for him. He shoved me away, and rather matter-of-factly said "Daddy stay there. Poppet will get a story." Then he turned to get the book, but quickly came back to me and added "I'll be back at half-past-four. See you soon." And off he went to get his favourite book (which, as ever, is about excavators and other large machines). 

That was just one of those amusing moments you get with a young child. But another 'time' incident was rather more interesting. Karen was out, and Benjamin was somewhat upset over her absence. I was trying to reassure him that she would be back. "She'll come home at about half-past-eight". Benjamin looked at the clock longingly and said "Daddy change the clock so it's time for Mummy to come home". 

Now there's a thought! Wouldn't it be great if we could just manipulate time by turning the hands of the clock? So, somehow, if we changed what the clock says, then the time of day would actually change. Does Benjamin think that this is how it actually is? Maybe. It would be useful if it were true - extended weekends, short work days; one could cut hours of a plane trip to Europe by taking the clock with you. 

While it might work for science fiction writers, unfortunately that's not how time works for us. While the clock and time are intimately linked, it is time that controls the clock, not the other way around. We are stuck with progressing through time at the rate of one second every second. 

That makes time a rather strange concept from a physics perspective. Unlike space, where we are free, more or less, to move to any point in it, we don't have that option with time. We can only move forward in it, and only move forward at the same rate - one second every second. The past is forever behind us; and the future is always unknown. Physicists call this the 'Arrow of Time'. It points one way: forward. 

Special Relativity makes it more interesting still. The way time works for you may not be the same as it works for me. If I were to get on a Really Fast Spaceship and travel close to the speed of light for a while, then return to earth, I would be noticeably younger than my identical twin brother. (I actually DO have an identical twin, by the way.)  Not only would the clock in my spaceship be telling me less time had past, I would have actually aged less. From my point of view, I might have been gone for two weeks; from yours, I might have been gone ten years. But even so, each of us will still have perceived time as travelling at one second every second. Forward. 

What about real time-travel - going back in time. Just maybe physics permits this to happen. That's in the realms of General Relativity and Quantum Gravity and involves some really big masses indeed. Matt Visser's work at Wellington might give us some pointers here. But the summary of it is: Don't expect a such a time-machine to be built in Benjamin's lifetime, even if he does prolong it indefinitely take the battery out of our lounge clock.

 

 

 

 

 

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