Introduction

Usain Bolt is currently the fastest sprinter in the world. His two world records in last year’s athletics world championships were astonishing even for his standards, but what next? Bolt has often talked about reverting to competing in his original event of the 400 metres, and has already competed in a 400 metre race this year. If he focuses his training sufficiently, he may be able to top Michael Johnson’s 400 metre world record (43.18 seconds), which has stood for 11 years. Bolt has also told his coach that he would love to try his hand at the long jump and thinks that he would be very good at it. Is this likely? Could Bolt be any good at the long jump, and will we see any new world records?

Long jump

Just like the sprint events, explosive power is an essential attribute if you are trying to learn how to jump higher, competitors sprint along a 40 metre runway before jumping in to a sand pit. A jump is only counted if the athlete takes off behind a foul line, which is marked out by a board. The distance of the jump is measured from this foul line to the nearest mark made in the sand pit. It’s not hard to see why Usain Bolt might be successful at this event, in theory the faster you can take off the further you will be able to jump, with Usain Bolt’s credentials you’d expect more than a modicum of success.

According to data collected at the 2009 Berlin World Championships, Bolt managed to top 12ms-1 at the 40 metre mark of the 100 m final (graph), approximately 15% greater than his average velocity. If Usain Bolt was able to maintain this velocity during take-off, we can calculate maximum jump distance using elementary kinematics. With an optimum take-off angle of 45° and ignoring air resistance jump distance D, is given by:

where v is the initial take off velocity and g is the acceleration due to gravity. With v = 12ms-1, and g = 9.81ms-2, D is a massive 14.68 metres. This is 64% more than the world record of 8.95 metres held by Mike Powell and set in 1991. However, this equation does not take in to account the velocity lost during take off, the height difference between take-off and landing, and the optimum take off angle because of this change in height. However it does demonstrate that the faster the take off velocity v, the longer the distance of the long jump.

Modelling the long jump

A detailed long jump kinematic model will produce a more accurate estimation of Bolt’s potential long jump distance. Tan and Zumerchik have produced a model to relate horizontal running velocity to long jump distance (Kinematics of the Long Jump). This model takes in to account the energy lost when transferring horizontal velocity in to vertical velocity, as well as the change in height of the centre of mass of the long jumper. Geometries used by the model are shown below in the diagram:

Using the model and applying it to Usain Bolt, we find that with at 1.93 m high, constant ‘a’ (distance to centre of mass) will be 0.965 m and ‘b’, (distance to centre of mass at landing) will be approximated as 0.6 m. From Tan and Zumerchik’s model, the jump distance R is given by:

where ‘v 1 ‘ is the initial take off velocity, ‘α‘ is the take off angle, ‘h‘= a–b sinβ and ‘β‘ is landing angle, taken to be about 45°. The take off velocity ‘v 1 ‘ is the initial horizontal velocity ‘v 0 ‘ and the equation:

‘1-γ‘ is the efficiency of the velocity transition and is taken to be around 90%

Results

According to the model, the existing world record of 8.95 m set by Mike Powell in 1991, was set with an initial running speed of 11 ms-1, and a take off angle of 33.06°. This take off velocity is a full 1 ms-1 lower to what we expect Usain Bolt to achieve. Using a spread sheet optimisation function and the long jump model, Usain Bolt could achieve a long jump distance of 10.50 metres with a take off angle of 33.24°, his take off speed ‘v 1 ‘ would be 9.98ms-1.

Discussion

The model suggests Usain Bolt could beat the long jump world record by a whopping 1.55 metres. This is a massive margin, but nobody thought that the 100 metres World record would drop to as low as 9.58!

There have been many athletes in the past that competed in both the 100 metres and the long jump. Famous examples include Marian Jones with a long jump distance of 7.31 metre and 10.65 seconds in the 100 metres, Jesse Owens with a 8.13 metre long jump and 10.2 seconds in the 100 metres and finally Carl Lewis with a long jump of 8.91 meters and 9.86 seconds for the 100 metres.

In his time, it was estimated that Carl Lewis could have achieved a top running velocity of 11.5 ms-1. This is only 0.5 ms-1 down on what we expect Usain Bolt to achieve. Using this initial velocity and applying it to the long jump model, Lewis is predicted to have jumped 9.69 metres! However in the 1991 long jump final at the world championships, Lewis could only managed a jump of 8.91 metres, 4 centimetres shorter than Mike Powel and 78 centimetres shorter than what the model predicts. There could be many reasons for this. For example, the repeated sprints down the long jump track could have tired Lewis out reducing his horizontal velocity, or his long jump technique was not as efficient as Mike Powel who is thought to have had a slower horizontal velocity.

It seems that running extremely fast down a long jump runway would be the easy bit for Bolt. However transferring his translational kinetic energy in to an equally impressive long-jump-distance requires a specific long jumper’s technique. If Bolt is coached correctly and learns an efficient long jumping technique, we may see a new world record. Even if Usain did not master the most efficient long jumping technique and he wastes 20% of his original energy rather than the initial estimate of 10%, his predicted long jump distance will still be 9.46 metres.

So could Usain Bolt actually break the world record in the long jump?

Yes is the short answer. Usain Bolt seems to already have many of the attributes of a world class long jumper. However it takes a specialist to compete at the long jump, and an efficient jumping technique is required. It’s just a matter of whether Bolt can learn to jump efficiently and turn his record running speed and kinetic energy in to long jump distance. Bolt is a formidable athlete, it would be no big surprise if he manages to do this at some point in his career. So watch this space, Usain Bolt – fastest man in the world could be accompanied with Usain Bolt – longest long jumper in the world.

Evidence, does faster sprinters mean longer long jumpers?

Evidence for this can be found in athletics records of the long jump and 100 metre sprint. Long jump and 100 metre performances have increases since the start of long athletic competitions. Figure 1 shows the average of the top 25 male long jump and 100 metre performances since 1891. There is a clear rise in long jump distances and drop in 100 metre performance times.

If we take the average of the top 25 long jump performances each year for male and females athletes and plot these against the average velocity of the top 25 100 metre performance we get the second graph shown below. From this graph we can see that there is a clear linear trend. An increase in average velocity in the 100 metres means and increase in long jump performance. It’s important to remember that in most cases, completely different sets of athletes are being compared.

How does this tie in with the simplest long jump model?

Now if we apply the simplest parabolic flight model to the average velocity in the 100 metres each year, we can see how average velocities in the 100 metres relates to long jump distance. Remember to note however the average velocity upon completing the 100 metres is lower than the maximal velocity that can be achieved. If we look back early, Usain Bolt’s maximum velocity was approximately 15% lower than his average velocity. We also need to consider the launch velocity V 1 which in the case for Usain Bolt is approx 17% lower than his maximal speed. This means using a quick approximation the take off velocity v 1 will be 0.9545 x average velocity (1.15 x 0.83). Plotting this new take off velocity calculated from average velocity in the 100 metres produces the graph:

The three coloured lines are the simple parabolic flight model for three different value of ‘α‘ take off angle. We can see that long jump performance data closely matches this simple model when α = 35°. This angle ties in closely to the optimum angle (33.84°) calculated for Usain Bolt using the more detailed model of the long jump.

The evidence presented here suggests that there is a strong link between sprinting velocities and long jump distance, and as average velocities in the 100 metres event have increase over the years, long jump has increased inline with these increases. The improvement in human athletic performance demonstrated is not fully understood, or meaningfully quantified. Look out for future blog posts on the topic of improvements in human athletic performance.

Leon Foster