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A decathlon of questions on the physics of sport

16 Jul 2021 Laura Hiscott
Taken from the July 2021 issue of Physics World where it first appeared under the headline "Sporting chance". Members of the Institute of Physics can enjoy the full issue via the Physics World app.

With the Olympic Games due to get under way in Tokyo this month, Laura Hiscott brings you 10 physics-related sporting questions about the 10 different events in the decathlon – one of the highlights of the athletics programme. You can find the answers at the bottom of this page, or listen to Matin Durrani and Laura Hiscott discussing them on the latest episode of the Physics World Weekly podcast.

Photos of high jump, hurdles and discus

Day 1

100 m

In 1977 fully automatic timing, as opposed to timing done by people with stopwatches, became mandatory for world records in the 100 m sprint. Immediately after this change, average recorded times of sprinters increased slightly, before decreasing again. How did automatic timing cause this single stepwise increase?

Long jump

Many of the best long-jumpers in the world appear to continue running in the air as they cycle their legs for a few steps after take-off, in a technique called the hitch kick. What is the purpose of this motion?

Shot put

The women’s shot put has a mass of about 4 kg, but the volume varies slightly. If it can be made of solid iron or solid brass, what is the range of possible diameters it could have?

High jump

In the high jump, athletes traditionally keep their body upright as they kick their legs over the bar. But at the 1968 Olympic Games in Mexico City, American high-jumper Dick Fosbury won gold using a new technique he had developed. Now called the Fosbury flop, it involves slinking backwards over the bar and landing on your back. What physical principle does the Fosbury flop use to help an athlete clear a higher bar?

400 m

In the 400 m race, the starting line is staggered across lanes to ensure that all athletes have the same distance to run while staying in their lanes. Why does World Athletics say that the number of lanes for a standard track should be no more than nine?

Day 2

110 m hurdles

The men’s 110 m hurdle event has 10 hurdles spaced at 9.14 m from one another. The take-off foot should touch the ground at 2.1–2.2 m in front of each hurdle, and the athlete normally lands about 1 m from the hurdle. Most athletes take three steps between hurdles (not including the hurdle jump). About how long should each stride be?

Discus

The theoretical optimal angle for throwing an object as far as possible is 45 degrees to the ground. However, most athletes have an optimal angle slightly smaller. Why is this?

Pole vault

For an athlete with a centre of mass 1 m above the ground, who can run at 10 m/s, what is the theoretical limit to the pole vault height they can clear? And why is the pole vault world record slightly above this?

Javelin

In 1986 the men’s javelin was redesigned so that the centre of mass moved 4 cm closer to the tip. The women’s javelin was similarly redesigned in 1999. Which two problems prompted this redesign, and how did it solve them?

1500 m

The 2016 Olympic Games in Rio de Janeiro saw the slowest 1500 m Olympic event since 1932. Why is this event getting slower?

Answers

100 m: People with stopwatches tend to underestimate sprinters’ times. This is because they start measuring slightly after the runners have taken off, due to non-zero reaction times. Their reaction times to the runners crossing the finish line are not as long, because they are watching the runners and can anticipate when they will get to the end.

Long jump: Long-jumpers generate some angular momentum as they take off, which would cause their body to rotate forwards so that they are leading with their face, making it difficult to land on their feet. By cycling their legs, they take care of the angular momentum while keeping their body upright so that they can land feet-first. This is similar to how you might instinctively swing your arms when you lose balance. Not all long-jumpers use this technique, though. Some use the “hang-style” technique, in which they kick their legs forward after take-off, so that they are horizontal to the ground. They also fold their body forward over their legs. This lengthens their moment of inertia, so that the angular momentum generated causes their body to rotate less.

Shot put: Using density of iron as 7.86 g/cm3 the diameter is 9.91 cm. Using density of brass as 8.73 g/cm3 the diameter is 9.56 cm. The official range in the regulations is given as 95-110 mm. The shot put is not always just one material. It is sometimes made of a smaller lead weight in a metal casing of lower density, leading to a wider range of possible diameters.

High jump: When an athlete slinks over the bar using the Fosbury flop technique, rather than going over it with their body upright, there is no point in time at which their whole body is above the bar. In fact, there is no point in time at which an athlete’s centre of mass goes above the bar – their centre of mass actually goes below the bar. This means that they have to generate less energy to clear the bar than they would if they went over it with their body upright, which would require them to raise their centre of mass higher. Therefore, at the maximum energy they can generate, they can clear a higher bar with the Fosbury flop technique.

400 m: The curved part of the track is sharper on the inside lane than on the outside lane, due to the increasing radius of curvature from the inside to the outside. It is harder to run around a sharper bend, so having a gentler curve may confer an advantage on the runner in the outside lane. However, World Athletics (formerly the IAAF) considers this difference to be mostly negligible, only becoming significant if there are more than nine lanes in the track.

110 m hurdles: The stride length should be about 2 m per step. The spacing of hurdles means that “rhythmic running”, with regular stride lengths, is more important in hurdle events than in flat races.

Discus: The basic model of an object being thrown from ground level with a given force gives 45° as the optimal angle to maximise the distance it will travel before hitting the ground. However, the real-world scenario is more complex. Athletes do not throw from the ground level, but a little above it depending on their height. The object can therefore be considered to be starting at a different point in the parabolic model. You can extrapolate the object’s motion backwards behind the athlete to imagine it starting at ground level. Imagining it taking off from the ground at 45°, you find that by the time it reaches the athlete’s hand, its angle to the horizontal would be less than this value. The athlete should therefore throw it at this lower angle to maximise its distance. There are also biomechanical factors, as athletes may be able to generate a greater force by throwing at an angle closer to the horizontal, due to how our muscles are arranged anatomically. Finally, air resistance also plays a role in real-world scenarios. Since it slows down the object’s horizontal motion and reduces the distance it travels, throwing it with a greater horizontal component may help to counteract this.

Pole vault: Equating kinetic energy with gravitational potential energy, the change in height is about 5.1 m. Adding the initial centre of mass being at about 1 m, this gives a theoretical maximum of about 6.1 m. The world record is currently 6.18 m. The extra height could be because, similarly to high-jumpers, pole-vaulters often use a technique where their centre of mass goes under the bar. Also, as the pole straightens, the athlete pushes off the end of it, putting more energy into the system, which gets added to the kinetic energy they generated in the run-up.

Javelin: The two problems were that 1) athletes were throwing the javelins increasingly far, which started to become dangerous as the javelins could go beyond the field into the audience, and 2) the javelin would often land flat on the ground rather than pointing down into the ground, leading to ambiguity around what was a valid, qualifying throw. When the centre of mass was moved towards the point of the javelin, the javelin’s trajectory changed, so that it started to point downwards sooner in its motion. This reduced the distances it could be thrown to a safer level and reduced the number of instances where it landed flat instead of point-down.

1500 m: The 1500 m is a very tactical race. Rather than trying to run it in the shortest time they can, runners often try to conserve their energy by running slowly throughout the race, staying behind the person in the lead. They then try to win by sprinting at the end. However, this means that no one wants to strike out and be the person in the lead, because that would mean wasting energy that the other athletes conserve, and would probably lead to being overtaken later. So none of the athletes end up running very fast because none of them wants to be in the lead. This leads to an example of a “Nash equilibrium”, which is a concept in game theory where no participant has anything to gain by changing their strategy. In some competitions, pacesetters run parts of the race with the athletes, which encourages them to run faster rather than running strategically. However, no pacesetters run in the Olympic Games, and athletes would usually rather win an Olympic gold medal than beat their own personal best.

  • Many thanks to Steve Haake, professor of sports engineering at Sheffield Hallam University for checking these answers for accuracy
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