Plus Blog

August 10, 2012

The beautiful game has been saved for last at London 2012, with the men's gold medal match taking place on Saturday, the penultimate day of the Games. There are some important questions to ponder while we sit tight in anticipation for the final match. What's the best strategy for taking a penalty kick? When is it worth committing a professional foul? And when is a goal not a goal? Find out about all this and more with our collection of football articles.

When is a goal not a goal? — Remember Frank Lampard's disallowed goal in the 2010 World Cup match against Germany? The ball hit the crossbar, landed well behind the line but then bounced out again. And it all happened too quickly for the ref to spot it was a goal. But do how these kind of (non)-goals happen and what can we do about it?

A fly walks round a football — What makes a perfect football? Find out why the ball's surface is the most prized research goal in ball design.

Eye on the ball — Why the 2006 World Cup ball caused controversy.

Blast it like Beckham? — What tactics should a soccer player use when taking a penalty kick? And what can the goalkeeper do to foil his plans? This article uses game theory to find the answers.

Formulaic football — When's the best time to make a substitution in a football match?

Understanding uncertainty: How psychic was Paul — How likely is it that Paul the octopus' accurate predictions for the 2010 World Cup were really due to psychic powers?

Mathematicians rival octopus in World Cup final prediction — Is maths better than Paul the psychic octopus?

The luck of the draw — How a mathematical improbability turned one day's football action into a once-in-a-lifetime quirk of fate.

On the ball — If your team scores first in a football match, how likely is it to win? And when is it worth committing a professional foul? This article shows us how to use probability to answer these and other questions, and explains the implications for the rules of the game.

Understanding uncertainty: Football crazy — Taking up a challenge from a BBC radio programme David Spiegelhalter and Yin-Lam Ng used their statistical finesse to predict the outcome of the last matches of the English Premier League — and they were 90% correct. Find out how they did it.

Power trip — How long do football managers' careers last and does talent matter? Surprisingly the answer may lie in mathematical power laws.

Understanding uncertainty: the Premiere League — Does a club's ranking in the Premiere League table reflect how good it really is? This article discusses the concepts of mean, variance, standard deviation and confidence intervals.

August 9, 2012

A dressage horse and rider performing the extended trot. Image: Chefsna.

It's a great day for individual dressage today with the Grand Prix freestyle test taking place in Greenwich Park. It's amazing how those horses can perform elegant and complicated movements without getting their legs in a muddle. Coming to think of it, it's amazing that they can even go through their innate gaits without getting their legs in a muddle, given that there's four of them and they are very long. And what about animals who've got even more legs? It turns out that it's all down to symmetry in the brain. To find out more, read Ian Stewart's Plus article Walk, trot, gallop.

One area in which humans definitely have the edge over horses is ball games. Volleyball is a particularly exciting one and we're looking forward to the semi-finals today and tomorrow and the final on the weekend. As in tennis, spin's the thing in this game. But why does putting a spin on a ball make such a difference to its path through the air? John D. Barrow, mathematician, cosmologist and prolific popular science writer, explains:

One of the key skills for volleyball players to master is to serve the ball with a top spin so that it will drop down more quickly than the receiver of the serve might expect. This is possible because a spinning ball will follow a curved trajectory when a non-spinning ball will not. The reason for the swerve can be understood by looking at the air flowing past the ball. When the air flow impinges on the ball the flow lines are pushed together, so the pressure drops and the speed of the air passing the surface of the ball increases.

Spinning ball

Air flow around a spinning ball.

If the ball is spinning then the flow lines of the air near the surface of the ball are significantly altered. In the picture on the left you can see what happens if the ball has been given a clock-wise spin in addition to its straight line motion at speed V from left to right. The non spinning motion is just as if the ball were stationary with air going past its surface at speed V. What is the effect of the spin? At the top of the surface of the spinning ball the air speed very close to the surface of the ball is in the opposite direction to the on-coming air whereas at the bottom it is in the same direction. This means that the net speed of the air near the top of the ball is less than that round the bottom. So the pressure on the ball is greater at the top than the bottom and there is a net downward force, F. The orientation of the picture shows how a ball served to the right with top-spin will always swerve downwards. This will reduce the time that it spends in the air and give the receiver less time to respond and return the ball.

You can read more about the effect of spin on the trajectory of a ball in our article Spinning the perfect serve.

August 8, 2012
A table tennis player

What is your chance of winning at table tennis?

Today the men's table tennis teams will be battling for gold. Table tennis first became an Olympic sport in 1988, but changed its scoring system in 2001 to make matches more exciting for spectators. But how do the old and new scoring systems compare in terms of favouring skill versus luck? Find out in our article Ping-Pong is coming home.

And while we're on scoring systems, there's a video of a lecture given by John D. Barrow on our sister site Maths and sport: Countdown to the Games, which looks at them in detail.

August 7, 2012

Sir Chris Hoy leads the GB Cycling Team during the official opening of the Velodrome (Photograph by David Poultney)

It's the very last medal day for track cycling! If you've been watching, you'll have noticed just how steeply the elegant wooden cycling track is banked in the turns. Echoing this shape is the sweeping curved roof of the Velodrome (which hit the headlines in its own right during the torrential rain this weekend) topping the beautiful cedar clad exterior. To find out why and how the track and its buildings came to have the shapes they do, read the Plus articles Leaning into 2012 and How the velodrome found its form.

August 6, 2012

Yesterday was a great day for badminton with gold medals being awarded in the men's singles and doubles. This got us thinking about shuttlecocks. They are not like balls at all and this means that they don't behave like balls either. John D. Barrow, mathematician, cosmologist and prolific popular science writer, explains.

Shuttlecocks used for badminton are not like other projectiles found in sports. They are extremely asymmetrical, with a conical skirt about 6cm long and 6cm across that is attached to a cork of higher density at the narrow end of the cone. However, when the shuttlecock is hit it will quickly flip over so that the cork is leading because its pressure centre if different to its centre of mass. This will ensure that it is always approaching your opponent the right way round for her to hit it back.

The effect of hitting the shuttlecock is strange. When you hit a tennis ball or a cricket ball with a racquet or bat it goes further the harder you hit it. But no matter how hard you hit the shuttlecock it won't go much further than about 6 or 7 metres.

Trajectory of a shuttlecock

The trajectory of a shuttlecock, moving from left to right: it falls steeper than it rises.

The motion of the shuttlecock obeys Newton's laws of motion. Its acceleration is governed by the downward force of gravity and a drag force from the air that is proportional to the square of the shuttlecock's speed through the air. When it is first struck the shuttlecock is moving at its fastest and the drag force is therefore much bigger than gravity. So it moves upwards in a straight line, at an angle determined by the direction of impact from the racquet, gradually being decelerated by gravity. Eventually it is going so slowly that the force of gravity is comparable to the air drag force and the trajectory reaches a maximum height and curves downwards towards the ground. Gravity is now speeding it up and it quickly reaches a speed, termed the terminal speed, where the opposing forces of gravity (downwards) and air drag (upwards) become equal. There is now no net force on the shuttlecock and it moves downwards at this constant terminal speed, without experiencing any acceleration or deceleration. The overall trajectory doesn't look like a parabola: the shuttlecock falls steeper than it rises.

The terminal speed does not depend on the initial launch speed of the shuttlecock. It is determined by the strength of gravity, air density, the size and mass of the shuttlecock, and its smoothness. As a result it is these unchanging properties that fix the distance that the shuttlecock will reach when struck hard. Hitting it even harder can't make it go any further than these properties dictate.

August 2, 2012
Usain Bolt

Image: Jmex60.

Usain Bolt is determined to become a legend this weekend, by running the 100m in 9.4 seconds. But what does mathematics have to say about this quest? What is the ultimate limit which no runner can possibly surpass? How can Bolt improve his record significantly without improving his speed? What about the effects of wind assistance, timing accuracy, and altitude on sprint times? Find out about all this and more in the Plus article No limits for Usain and in this video of a lecture given by John D. Barrow and featured on our sister site Maths and sport: Countdown to the Games.

The weekend will also be a great one for Olympic tennis so you should rehearse the scoring rules:

You win a game if you score 4 points before your opponent scores 3 points. Or, if you both score 3 points at some stage you win if you manage to score 2 points in a row after the 3-all stage before your opponent does.

That's quite a mouthful and it turns out that the maths behind tennis can get tricky too. What is the secret to the perfect serve? Can you figure out the probability of winning a game when the probability of winning a point is 0.6? What's the chance of a tennis match taking on epic proportions like the 11-hour battle between John Isner and Nicolas Mahut at Wimbledon in 2011? Why does tennis use this comparatively complicated scoring system? And can you improve your chance of winning by choosing the right racket? Here are some answers to these questions, from Plus and from our sister site Maths and sport: Countdown to the Games:

Spinning the perfect serve — A new mathematical analysis of how to hit a winning server shows that spin is the thing. Perhaps there's still time for Murray's coach to include some maths in his preparations for the match today...

Any win for tennis? — Work out the probability of winning a game for a fixed probability of winning a point. This challenging activity is designed to be accessible to students of A-level maths and anyone else who likes puzzling over probabilities.

Anyone for tennis (and tennis and tennis ...)? — What's the chance of a tennis match taking on epic proportions like the 11 hour battle between John Isner and Nicolas Mahut at Wimbledon in 2011?

Final score — Why are there so many different scoring systems in operation in sport? This video of a lecture given by John D. Barrow looks at how structuring matches into a series of sets affects the relative roles of luck and skill in determining the winner of the contest. It also looks at issues surrounding scoring in table tennis and decathlon.

Making a racket: the science of tennis — While the players get most of the limelight, engineers, too, are working hard to produce the cutting-edge tennis rackets that guarantee record performances. Over recent decades new materials have made tennis rackets ever bigger, lighter and more powerful. So what kind of science goes into designing new rackets?

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