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Sunday, 27 May 2012

The Physics of Field Athletics: Why do the Shot, the Javelin, the Discus and the Hammer fly differently?

Shot-put, Javelin, Discus and Hammer-Throw all have the same basic concept. Get the projectile as far away from you as possible before it bounces. However, the distances and methods of throwing are very different for all of them. So, what is the physics behind them?

There are a few differences between the sports. The distance in shot-put is noticeably smaller than in the others. In all but javelin, the athlete spins before throwing. I am going to go through and look at the differences between the events. This week, we will talk about shot-put and javelin, and in a few weeks time we will look at discus and hammer-throw

Firstly, lets look at shot-put. The men's involves "putting" a 7.25 kg (16lb) "shot" from a small 2.135m (7.00 ft) diameter circle. The put is thrown with a sort of push, instead of conventional throwing, due to its weight. It is also the weight that leads to the substantially smaller distances that a shot-put athlete achieves. Unlike javelin, shot-put requires not only explosive arm and abdominal strength, but also an even more powerful leg push. Anyway, lets take a look at the current world record, done by Randy Barnes, back in 1990:

What is interesting about shot-put is just how much force Randy has to put in to the shot in order to achieve his impressive 23.12 m (75'10"). For this, we have to use the laws of trajectory. At the moment, we know the distance that the shot travels and that is about it. What we want to find out is the initial velocity when the shot leaves Randy's hand. Here is the equation we will use:

V0 is the initial velocity, R is distance the shot travels, g is gravitational acceleration towards the ground (around 9.81 ms2)  and θ is the angle at which the shot is putted.
We don't know the value of the angle, however what we can do is accept the angle to be 45˚. This is the angle that would mean he would have to put in the least effort. So lets work it out:


Now, force is shown by the expression:

Where F is the force, m is the mass of the object and a is its acceleration

We can also change this into:
v is the final velocity, u is the initial velocity and t is the time
The initial velocity when he is throwing is 0 ms and the final velocity is, as we worked out, 15.06 ms.  All we  have to do now is know the time it takes for this velocity change. From the video, we can see it takes about 1 second. Therefore:

To put this in perspective. The average person is around 70kg so if we plug in the numbers, and I won't show it because it is tedious, we find that he could throw a person a quarter of a meter. This doesn't sound like a large distance, until you consider that they are throwing a dead weight fully grown adult from a dead stop. It is pretty impressive!

So, now to javelin. There are a few things that set javelin apart from the other throwing sports. It is the only one to have one method that the athletes have to use. Another difference arises from this, the allowed throw has to be an overarm throw as you would throw a ball, and the athlete doesn't spin at all - completely unlike all the others. This means that almost all the power put into the throw was done only by the arm of the athlete, with a little bit of momentum gained from a 30m run up.
Back in 1984, the javelin had to be redesigned. This is mainly due to this absolutely incredible throw two years earlier:


So now lets take a look at the more recent world record:

So what are the main differences? The first thing you notice is that the first throw seems to glide, going further with what appears to be a shallower curve. Secondly, in the first video, the javelin seems flat almost the whole time of flight and lands nearly parallel to the ground, where as the second javelin tilts down more and embeds itself in the ground nearly perpendicularly:
Blue is the first video, red is the second.
It depends on the style of the athletes as to whether these lines overlap.

So what were these changes and why were they put in place? Well in the first javelin video, the javelin was thrown 104.8 meters. It was decided that the throws were getting so long that they would soon become dangerous as they outgrew the length of stadiums they were held in. Another problem was that the landings started to become shallower and shallower, and seeing as the javelin must land point first, there were disagreements over some throws. Therefore the decision was made to move the centre of gravity of the javelin 4 centimeters forward. Though this doesn't sound like a large change, it caused the javelin to dip considerably more, shortening the distance by around 10%.


We hope you enjoyed this post, it was certainly very interesting to research and write. If 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.


Ned Summers

Check out our last two posts:

The Physics of Gymnastics - The forces exerted on a gymnasts body during some routines are extreme, so how does it work?

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?

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.


Sunday, 13 May 2012

The Physics Of Cycling: Why Does A Velodrome's Sides Need To Be Banked?

Earlier this year, I was lucky enough to go to the new Olympic velodrome in London. I was shocked to see how steep the sides were. Cycling is a sport with a huge amount of variety. There are events based on sprinting, events based on endurance and even events like the Madison where there are people both going fast and going slow. So, how are velodromes designed to maximise the speed of riders in all categories?

There are two main things that lead to the need for banked sides at a velodrome. One of which is a simple property of curved movement, the other is an intrinsic property of bicycles themselves.

So lets start with the first one. We have all experienced a so called "Centrifugal" force. This is the force that, when you are going around a corner in a car or spinning on a children's merry-go-round that you find in a playground, seems to pull you outwards, pushing you into the car door or off the merry-go-round. In fact what you are feeling is not really a force pulling you outwards, it is instead the effect of the force pulling you inwards:

When you want to turn from going in the original direction, you have to induce the force towards the centre of the turn. You feel like you are being dragged away from the corner, because that is the original direction your body was going in and it doesn't want to change. This force does come into play when cyclists have to turn corners in a race and can lead to problems. When you turn on a flat surface, your change in direction relies purely on traction, or friction, that your tires exert on the road. The equation that shows how fast an object can go before it slides off the surface is this:
Which we can simplify to

Where v is velocity, r is the radius of the turn and g is the acceleration due to gravity. The weird u-like character is the Ancient Greek letter Mu, and it represents the coefficient of friction. Despite its complex name, the coefficient of friction is just a number that shows how well two substances grip each other. So lets work out how fast an Olympic cyclist could go around a corner if it was completely flat.

The radius of a turn on an Olympic velodrome is around 20 meters, the acceleration due to gravity is 10 meters per second squared, and the coefficient of friction between rubber and wood is roughly 0.7-0.9. So if we put that into the equation we get:

This roughly equates to 45.5 km/h. This means that a cyclist can only go at about 45.5 km/h around a corner without losing grip and possibly being dragged away from the inside of the track. Racing cyclists get to speeds as high as 80 km/h, which would cause problems on flat tracks. However, banking the corners alleviates the problem. Instead of getting pulled to the outside of the track, instead the cyclist is pulled into it:

The equation used to find the angle of the slope is a reasonably simple one:


Where v is again the maximum velocity of the bike and r is the radius of the curve, whilst the 0-like character is the Greek letter Theta, and just represents the angle. So lets work out the maximum velocity that the bike can go. The radius remains 20 meters and the acceleration due to gravity remains around 10 meters per second squared. The angle of an average 250 meter track is around 45 degrees:


Ok, so that isn't yet high enough. However, what we haven't accounted for is the fact that the cyclist leans into the corner, instead of solving for a speed, lets solve for the angle.


This means that for the cyclist not to slip the bike has to be quite close to the ground. The slope of the side helps the bike get this angle.

However, there is another property that requires a tilt in order for the bike to turn at maximum speed. This property is one that applies specifically to bikes and other objects with a similar configuration to bikes. This property is that the bicycle is an inverted pendulum. This means it pivots around a point which is below it's centre of gravity:

So what does this mean? Well the best way to think about it is to imagine that you are riding a bike. When you turn a corner there is one main change to when you are moving straight, you lean in. So why is this? Well it relates back to the pendulum idea. When the bike turns, the centre of gravity has to be over the inside of the turn. Imagine leaning away from the inside of the turn when on a bike, the result is inevitable, you will fall over. The angle that the bike has to lean at is expressed by the formula:

But wait, if we scroll up, that formula is very similar to the one we saw before. In fact, if we rearrange it, we get:

which is exactly what we used to find the angle needed so that the bike won't slide off the track. These two properties are in fact linked, and that means that solving one problem cannot make the other worse. Instead, when you solve one problem, you also solve the other. We already calculated the angle the cyclist has to be at, around 68˚. The slope of the track cannot be this steep, it just isn't practical. Any cyclists in a event like the Madison that need to go slowly will just slide down the sides. Instead the sides are usually around 45˚ and then the cyclist leans in order to make the turn.

There is one more reason a cyclist would prefer for the floor to be sloped, rather than them having to lean steeply into the corner. The leaning into the corner is both a factor that needs to be done before the bike can turn and also a way of turning the bike. The more a cyclist leans, the sharper they can turn. In fact, as a cyclist leans, they are turning the bike a little simply by leaning. The fact that the track slopes, means that it nearly turns the cyclist by itself. This means it is a lot easier for the cyclist to continue moving at speed, instead of having to slow down.

Finally, the reason the straights are also curved is simply so that it would not be as large a change between the angle of the straights and the angle of the turns.

In conclusion, the sides on a velodrome are sloped so that the cyclists won't slide of the track, so that they will not have to lean at extreme angles in order not to fall over and so that they can keep speed whilst turning.

Lastly, I think I should show you a fantastic piece of cycling, to finish off. I was lucky enough to be at the Melbourne velodrome when this happened. The whole race is very good, but watch from 4:00 to see the really amazing stuff. Our boy, Chris Hoy steals the gold beautifully:
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.

Ned Summers

Check out our last two posts:
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?
The Physics of Football/Soccer: What is "Curl"? - How do soccer players manage to change the trajectory of the ball as it flies through the air?

What are we posting about next:
The Physics of Gymnastics - The forces exerted on a gymnasts body during some routines are extreme, so how does it work?

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


Sunday, 6 May 2012

The Physics of Rugby: Forward Passes and Relativity

Hello, and welcome to The Aftermatter for another post in our series on sport, this time talking about Rugby. How can a ball thrown backwards travel forwards? Why there are so many forward passes after tackles? And why is Rugby a bit like Einstein's Theory of Relativity?



For those who don't know, in Rugby, only passes that are backwards or directly horizontal are permitted. Unlike in similar games, like American football, the ball is not allowed to be thrown forward. This is what gives Rugby such structured and tactical play, but it also presents certain problems with officials, and that is all to do with relative velocity.

Relative velocity in one dimension is pretty simple to understand. You are walking North at 4mph, when a jogger passes you heading North at 9mph. What is the velocity of the jogger relative to you? Since you are travelling in the same direction, you just subtract the velocities, giving him a velocity relative to you of 5mph North. If the jogger was running in the opposite direction to you, you would add the velocities so his relative velocity would be 4+9=13mph South.

In fact, every speed that we consider is in fact relative. When we are standing still, we generally say that we are moving at 0mph. However, that is only relative to the ground. Relative to the centre of the Earth,   you could be moving at up to 1,070mph (if you are at the Equator); relative to the sun, you are moving at over 67,000 mph; and relative to the centre of our galaxy, you are moving at nearly 500,000 mph. It's just that in real life   situations, we mostly measure speed or velocity relative to the ground we are standing on, but it is important to remember that it is only for our convenience that we use this as a way point; there is nothing absolute about velocity.

Now, the difference between velocity and speed is that velocity takes direction into account. For example, it is impossible to have "negative" speed: speed is a purely quantitative measure, just saying how much you are moving. Velocity is quantitative as well, but it is also qualitative: it says in which direction you're moving. For this reason you can have negative velocity: -4mph North is the same thing as saying 4mph South. For this reason, if you are moving in a curved trajectory, your speed is larger than your velocity. This is because velocity is measured with a straight line from where you started, not with the actual distance you have moved.

In maths, things with both magnitude and direction are called vectors. These are usually notated by letters in bold, for example, a, or b. A simple example is shown below.

The length of the line is proportional to the magnitude of the vector, and each vector has its own direction. Vectors can be multiplied by a number, so 2a would just be twice as long as a, in the same direction. However, it is important to note that however many times you multiply a, you will never get b, or any other vector that is not parallel to it. This is because multiplying by a will only change its magnitude, but never its direction.

Most of the time when a ball is passed in Rugby, there are two forces acting on it: the player throwing it backwards, and the momentum of the player (and therefore the ball) carrying it forward. We can represent these forces with vectors a and b as in the diagram below, where the blue squares are the players, and they are both running to the right of the screen.

The most important point to realize is that the ball will maintain its momentum from when it was being carried by the player (by Newton's First Law of Motion, also known as the Law of Inertia).

In order to calculate the overall, combined effect of these two forces on how the balls travel, we have to add the vectors, creating the new vector a + b. To do this, we simply move in the direction and magnitude of b, then in the direction and magnitude of a, as below.

As we can see here, despite the player passing the ball backwards, the ball has actually travelled forward! So is this technically an illegal forward pass?

The official rules state that a forward pass occurs when a player "throws or passes the ball towards their opponents' dead ball line." However, in the above examples, the player has thrown the ball backwards, but the ball itself has travelled forwards relative to the ground. In other words, the ball must be travelling backwards relative to the passer when the ball is released. The IRB has agreed to this definition.

Most rugby players or officials don't think about this, as if the players carry on their run, then it creates the illusion that the ball is travelling backwards anyway, but there are a couple of situations where the officials have clearly not thought about or remembered the effect of momentum, and some incorrect calls are made.

One example is when the players are unfortunate enough to be passing when next to one of the pitch's horizontal lines. The umpire can see much more clearly that the ball has travelled forward relative to the ground, and thus the illusion that the ball has travelled backwards, which the umpire usually relies upon, is destroyed. Thus many incorrect calls are given.

The other common error is when the passer is impeded shortly after making a pass.

In this case, when the passer is tackled shortly after making the pass, the ball ends up overtaking the passer, meaning that it is suddenly very noticeable that the ball has travelled forward, and the umpire stops play. However, the fact that the passer was stopped does not mean that he passed the ball forward, it was only momentum that carried it towards the opponents' end. This is why so many forward passes are given when offloading from a tackle.

Umpires need to be made more aware of these phenomena so they can enforce the laws with more consistency. Bryce Lawrence, in particular, fell foul of this in South Africa's 2011 World Cup game against Australia, with exactly this situation occurring at 3:48 in the following video. South Africa lost this game by only 2 points and went out of the World Cup, so this decision changed everything.


As well as in Rugby, relative velocity is very important in other areas of Physics. For example, I stated earlier that all velocities and speeds are relative. However, this was not quite true, there is one exception, and that is at the speed of light.

Einstein's Theory of Special Relativity shows, and relies on the fact that the speed of light is the same for any observer, regardless of their own velocity, when in a vacuum. This means that if you are travelling at a speed very close to the speed of light, and your friend next to you is stationary, then light will have exactly the same velocity relative to either of you. This seems nonsensical, but it has been proven conclusively in experiments, and has the remarkable result that the faster you are travelling, the slower time passes. In other words, Einstein proved that time itself is always relative. You can read more about this in our previous post on Special Relativity and its effect on time.

In conclusion, we've shown how relative velocity affects everything from Rugby to Einstein, and how a lack of knowledge of Physics can lead to some major mistakes from line judges, and confusion for spectators and players. Hopefully the Rugby authorities will continue to work hard to educate officials on these topics and maintain consistency.

We hope you enjoyed this post. 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 Football/Soccer: What is "Curl"? - How do soccer players manage to change the trajectory of the ball as it flies through the air?
The Physics of Cricket: What is 'Swing Bowling'? - The first instalment of our "The physics of sport" series. How does a bowler make a cricket ball swing? We assure you, it is very different to how soccer balls curl!

What are we posting about next:
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?

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



Theo Caplan