Let the game begins. From now on till up to a month, sleep pattern for some people may change due to the hooking up to the tele to watch the World Cup Brazil 2014. Who wouldn't, as it's one of the most celebrated sports tournament around the globe apart from the Olympic Games. And it happens every 4 years, that's how you can use all the 'reserved energy' you've been saving after the previous one at South Africa!
Anyway, let's go back a little bit into the history. Way back, to more than a decade ago into one of the recorded event in the world of football (or soccer, for you lot in the States). Probably the most-studied kick in football history was David Beckham’s free-kick goal in the England-Greece World Cup qualifiers in 2001. The kick left his foot, it was high enough to pass over the screen of defenders, and spinning enough on a vertical axis to curve toward the corner of the goal. It appeared to be aimed above the goal, but suddenly slowed down dramatically in flight and fell into the upper corner of the goal. How do you explain this through fluid dynamics point of view?
Well, let's begin with what the concept truly lies. The keywords here : the ball, the flow, the air. Those who took Fluid Mechanics would've known this as the flow of solid particles through fluid medium.
How can we analyze it? First, we need some of the important data, coz without it we can't deduce the observation quantitatively.
- The speed of the ball = 36 m/s (as reported by literatures)
- The distance of the kick from the goal = 27 m.
- Surrounding pressure = 1 atm (a typical atmospheric level as I presumed it's not on top of the mountain!)
- Surrounding temperature = unknown, but let's assume it was 25oC (or 298K).
Why do we need the pressure and temperature? Coz we wanna know the density and viscosity of air, which gives us:
- Density = 1.20 kg/m3
- Viscosity = 0.000018 kg/ms.
What else do we need? Oh, the 'properties' of the ball itself. As FIFA-approved standard, let's take the ball as having:
- Mass = 425 g (or 0.425 kg)
- Diameter = 22 cm (or 0.22 m)
So, do we have enough information? What else we can assume to make our analysis easier (as what engineers would do!)? Well, let's assume that the stitching on the ball, spin, gravity and wind that influence the speed and curvature of the ball's flight path are initially disregarded. Otherwise, this preliminary analysis will be neverending!
Now, we have almost all the information ready. What's next? Let's sketch how the situation would possibly look schematically. One word of precaution here: in actual situation, the ball would move in a curved projectile mode, hence the effect of gravity must be considered (as the initial statement was 'fell into the upper corner of the goal'). But in this preliminary analysis, I assumed it moved in a straight line horizontally.
Taking the 'control' area is surrounding the ball, as it moves very fast from Beck's foot towards the goal post, we can assume the buoyancy effect subjected onto the ball is very minimal (~ 0) in the vertical z direction. The affect of the gravity (i.e. weight, W) is also assumed to be negligible, as it was also in the vertical axis. Hence, the forces that involved during the kick is the force by Beck's (F) and the drag force due to the air that acted on the opposite direction (FD).
So, the equation would be: Force = Weight - Drag force - Bouyancy Force.
And eliminating the terms of W and FB:
Notice that the left hand side of the equation could be expanded to indicate the change in term of velocity. Why do we need to find the change in velocity? Coz that's what causes it to slow down and entered the goal.
Further expanding the equation, and integrate with respect to the boundaries involved, we will get this expression...
Now, before we solved to find the respective velocities, we gotta calculate the Reynolds number as we need to determine what would be the value of the drag coefficient (as you can see the term CD in the equation). At the beginning of the ball’s flight, the particle's Reynolds number is:
At this particular point, the Reynolds number is close to where there's a sudden change in the drag coefficient, as indicated by the diagram below. As the value is way below the limit of 0.1, for the purpose of evaluation we can assume CD ~ 0.1 for easy calculation.
Now, how to determine the change of velocity? If the ball slowed down, Re is lesser, therefore CD should be higher (based on the graph). Assume that this transition occur at halfway through the goal, therefore we need to find what would be the reduction in velocities.
For the first half,
Which means, the ball lost about 5% of its initial velocity.
For the second half, the ball is slowing down. But as Re ~ 501,000, therefore it doesn't follow our initial guess (of CD would be higher); thus we can still use CD ~ 0.1 at this point. Therefore,
Which means, the ball lost another ~7% of its velocity.
In total the velocity of the ball decreases ~12% from the initial, which is considered appreciably a lot when it comes to velocity reduction! But of course, the actual speed of the ball once it reached the goal was much lesser than what we calculate here as the stitching on the ball, spin, gravity and wind effect all contribute towards the slowing down of the ball (which we initially assumed to be disregarded) - which could probably reach 20% reduction. Massive? of course.
The outcome? GOAL!
Anyway. The footballs used in the matches must qualify the approved criteria set by FIFA. Ever wonder what are the testing procedures that the manufacturers must abide in order for the ball to pass before it can enter the field? These are the typical quality they must follow: