### 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