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Sunday, 20 May 2012

The Physics of Gymnastics: Vaulting and Newton's Laws of Motion

In gymnastics, gymnasts learn to manipulate their bodies in order to take advantage of Newton's Laws of Motion so that they can pull off spectacular tricks. So, how do they do it?

The event we will look at today is the vault. In the vault, a competitor tries to execute as many flips, twists, and spins as possible, whilst maintaining good execution and a balanced landing, after jumping off a springboard and over the horse, as shown below.



In order to complete as many flips and spins in the air, two things are important: air time, and rotational velocity. More air time allows more time to rotate, and increased rotational velocity makes the time required to complete each manoeuvre shorter.

The inevitable force that will end the jump is, of course, gravity. Gravity is a downwards force, so in order to delay hitting the ground and the jump ending, the gymnast has to exert an upwards force.

Now, let’s just think about what force is. A force is any influence that changes the motion of a body. So any force that is applied, must affect the velocity (to some degree) of the body it is acting on. Isaac Newton came up with a formula that represented this, known as Newton’s Second Law.

F=ma

Where F is the force, m is the mass of the body it is acting upon, and a is the acceleration of the body it is acting on.

In other words, the force is directly proportional to the acceleration it will cause. It is important to note that the technical definition of acceleration is not the same as the colloquial one. Technically, acceleration is a change in velocity, not speed. Therefore, an object can accelerate without getting any faster or slower, just by changing direction! By running around in a circle you are actually constantly accelerating.

The first force that the gymnast produces is in their run up, when they are trying to reach a maximum velocity before reaching the springboard. They then jump onto the springboard, and are launched into the air. Why does this happen?

This is due to Newton’s Third Law of motion. This states that all actions have an equal and opposite reaction. People utilise this all the time, for example when jumping. When we jump, we exert a force upon the ground beneath our feet, which gives us an equal force and propels us into the air. However it is not just us that moves, the Earth accelerates as well! However, since Newton’s First Law means that mass is inversely proportional to acceleration, the Earth’s huge mass means that we accelerate 75 billion billion times faster than the Earth, rendering this negligible in everyday situations.

Back to gymnastics: When the gymnast jumps onto the springboard, they exert a downward force, causing the spring to compress. The spring then returns to its original shape and exerts an equal, upward force on the gymnast. This is the same principle behind trampolines, and all elastic materials.

The gymnast then pushes off the horse using their hands, again exerting a downwards force and having an equal upward force exerted to them by the horse.

So, how can a gymnast increase the downward forces upon the springboard and the horse? One obvious answer is to run faster in the run up. A higher velocity means a higher momentum, which allows the gymnast to exert a greater downward force upon the springboard, which launches them further into the air. This then increases their downward momentum when landing on the horse, meaning there will be an even greater upward force.

But there is another method of increasing the upward force, which is to adjust the angle at which you land on the springboard. When running, you having a very high forward momentum, some of which you convert to upwards momentum after landing on the board, as in the diagram below.



Not much of the sideways momentum (as we look at it on the diagram) is converted into upwards momentum. However, if the gymnast hit the springboard at a more vertical angle, then the resultant force would propel them further into the air.



But how would a gymnast manage to come from this more vertical angle?

A Russian gymnast named Natalia Yurchenko came up with the answer in the 1980’s, now known as the “Yurchenko vault”. By doing what is known as a ‘round-off’ before reaching the spring, essentially two consecutive half-flips, she greatly increased the downwards momentum when landing on the springboard, and therefore the upwards momentum when springing off. Here is a video of Beth Tweddle demonstrating this technique.



The Yurchenko has another advantage; it provides an extra chance for the gymnast to increase their angular momentum. Angular momentum is defined as the product of the body's mass, linear velocity, and distance from the axis of rotation. This can be shown in the following formula:
L= r × m × v
Where L is angular momentum,
 r is the distance of the body of mass from the axis of rotation,
 m is the mass of the body
and v is the body's velocity.

Once the gymnast has sprung off the horse, they cannot gain any angular momentum, due to the Law of Conservation of Momentum (which comes from Newton's First Law: objects at rest will remain at rest and objects in motion will remain in motion unless an outside force acts upon them). The greater the angular momentum, the more potential for flips the gymnast has, so this is extremely valuable. The gymnast gains angular momentum by pushing off from a surface at an angle. In a regular vault, the gymnast only has two opportunities to do this, on the springboard and on the vault, but in a Yurchenko vault, they have four: jumping with their feet in the run up, springing with their hands on the run up, with their feet on the springboard and finally with their hands off the horse. 

Even after a gymnast has gained all their angular momentum, they can still work to control the speed of their flips. They can do this by changing the value of r in the equation above. Since neither the angular momentum nor the mass of the gymnast will change, lowering r will lead to the gymnast's velocity increasing. So how do we change this value?

When doing a flip, the "axis of rotation" is an imaginary line through the the centre of the gymnast, around the navel. r is the average distance of any given gram of your body from this line. When your body is stretched out, the value of r is quite high, as much of your body, particularly your head, is very far away from the centre of your body. However, if your body is curled up into a ball, the value of r will be very low, as almost all of your mass is very close to the centre of your body. The lower r is, the faster the gymnast will spin, which is why they often curl up into a tight ball in mid air to achieve more flips. The gymnast can then extend their body to slow their rotational speed to make sure that they land smoothly on their feet.

In conclusion, all three of Newton's Laws of Motion are needed to describe a vault, and gymnasts must constantly fight to increase both their upwards and angular momentum in order to stand a chance in this lightning-paced sport. Gymnasts train for hundreds of hours perfecting the most powerful runs, the strongest pushes, and the most efficient air movements, all in pursuit of the most beautiful jump. And, what's more, they have to make it look easy.

We hope you enjoyed this post, it was certainly very interesting to research and write. If you you want to get in touch you can follow and mention us on twitter, @theaftermatter, email us at contactus@theaftermatter.com or search "The Aftermatter"on Facebook.

Theo Caplan

Check out our last two posts:
The Physics Of Cycling: Why Does A Velodrome's Sides Need To Be Banked? - The sides of a velodrome are incredibly steep, so why is this?
The Physics of Rugby: Forward Passes and Relativity - How can a backwards pass look like they are going forward, and why is it so difficult for referees to judge?


What are we posting about next:
The Physics of Field Athletics: Why do the shot-put, the javelin, the discus and the hammer fly differently? - All these events have the same aspects, but they all get different distances. How does this work? 

If you have ideas for posts we would love to heard them. Contact details are above.


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