Zlatan Ibrah Rocket Science
‘Zlatan Ibrahimović! I want to go and give you a man hug!’
These were the words of commentator Stan Collymore after he witnessed the gigantic Swede rotate his body vertically through 180 degrees, meet the ball in a bicycle kick and lob it over Joe Hart’s head from over 25 metres. Seconds before, Collymore was calmly describing the game as a “worthwhile exercise” as England and Sweden looked forward to the World Cup qualifiers. Suddenly he exclaimed, “O my God, an insane goal! I’ve just seen the most insane goal I’ve ever seen on a football pitch!” Collymore’s voice then almost cracked as he declared his desire to hug Zlatan.
It was a spectacular goal. The bicycle kick is one of the most celebrated ways of scoring in football because it involves such a high degree of co-ordination, anticipation, accuracy and timing. The player has to turn upside down at high speed, follow the ball and meet it perfectly. For the ‘standard’ bicycle kick executed inside the box, like Wayne Rooney’s amazing goal against Manchester City, the player needs to make contact with the ball with the top of the boot. This ensures that the ball will travel downward towards the goal. Zlatan was well outside the penalty area, so he had to use the tip of his boot to arc the ball over keeper Joe Hart and the defence. His ‘bicycle lob’ was a whole new type of shot.
The execution may be difficult, but the physics of Zlatan’s bicycle lob is relatively straightforward. The main force involved is gravity, and the path of the ball can be worked out with the aid of Newton’s equations of motion. Assuming no air resistance, the ball will follow a trajectory similar to that shown in Figure 5.1 above. Like all such trajectories, its shape is a parabola I’m assuming that the velocity of the ball towards the goal is constant. The downward velocity of the ball increases over time as gravity causes the ball to accelerate towards the ground. Gravity provides a constant downward acceleration, so the ball’s initial upward velocity is positive, but decreases by an equal amount at every point in time. It is zero when the ball is at its maximum height and then becomes negative as the ball falls. As a result, the ball’s trajectory is symmetric around the maximum, and it follows the same path down as it does up.
Following the path of this parabola, the whole thing looks pretty simple. It is just a case of launching the ball at the correct angle and speed, then gravity will take over and the ball will land in the back of the net. But the problem is that the relationship between the angle of launch and where the ball ends up is far from straightforward. Figure 5.2 below is a plot of six different shots made at different launch angles, two of which end up in the back of the net and four of which miss.
All these shots had the same initial speed, 17 metres per second, but the outcomes are very different. Whether the ball lands short, goes in or flies over depends on a complex relationship between speed and angle. We can work out this relationship by solving the equations of motion. Showing how the ball moves with time is a task you might be asked to do in secondary-school maths. To solve for whether the ball goes in the goal, we need to rearrange the terms to find a condition for the ball to go under the bar, but not hit the ground before it goes in. This isn’t difficult, but it does require a few mathematical steps, shown below.
Figure 5.3 below shows the combinations of angles and speeds that result in the ball reaching the goal. Think about how Zlatan might choose a point on this plot. For example, he could kick the ball at 25 metres per second at 40 degrees, in which case the it will fly over the bar. Or he might kick it at 15 metres per second at 30 degree, but then it will bounce just in front of the goal. If, as we saw in Figure 5.1, he hits it at 16 metres per second at then it will go in. The goal-bound combinations occupy just a narrow sliver through the space of all possible angles and speeds. If the ball is hit too hard it will go over the bar; if it leaves the foot at too large or too small an angle it will bounce and be cleared by the defence. The weight on the ball has to be just right. The sliver is formed in a way that makes predicting what will happen without mathematics very difficult. For example, a ball struck at an angle of 19 degrees at 20 metres per second will go in, but just a bit harder than that and it will go over the bar. However, if the ball is hit at the same speed but at 65 degrees, it will go up very high, come down and drop into the net. This is a difficult shot to make, since even a small increase in angle or decrease in speed will cause the ball to bounce before reaching the goal.
So how did Zlatan get it right? Well, there was some degree of luck in the goal. Hart found himself outside his box and didn’t head the ball away far enough, so Zlatan was in the right place at the right time. But when he got the chance, he weighted the ball perfectly. I estimate, from my repeated viewing of the footage (with commentary on), that the ball left Zlatan’s foot at somewhere between 16 metres per second at an angle of 40 degrees (see Figure 5.1 above). By choosing this slower speed, Zlatan left more leeway for error in the angle. Any angle between 30 and 50 degrees would have put the ball in the net. If he had hit the ball harder, for example at 20 metres per second, the error margin would have been much smaller. Even when upside down, Zlatan minimised the probability of making a mistake.
The bicycle lob has relatively simple aerodynamics. The main force is gravity, and the equations are the same as those you learned in school physics classes. Zlatan applies an initial upward force, and gravity provides downward acceleration. However, it’s not quite as simple as that. In my calculations above I made things easier by ignoring the other forces at play here. As the ball flies through the air, the resulting drag slows it down, and Zlatan also applies spin so that the ball rotates as it drops into the net. There is a lot to think about when modelling the motion of a ball in flight.
Luckily, a bunch of rocket scientists are on the case. NASA runs a whole research programme dedicated to ball aerodynamics. They have even created an online shot simulator, where you can enter the position, direction, forces and spin on a ball and calculate whether or not it will hit the target. I don’t quite have the resources available at NASA, but I did add drag due to air resistance to my own Zlatan bicycle lob simulator. One simulated goal is shown below.
Air resistance is significant: it causes the ball to fall at an angle that is steeper than the launch angle. Figure 5.4 shows the ball being launched at 27 degrees but when it reached the goal its angle to the ground was closer to 80 degrees. The fact that the ball is falling more steeply, means that the slither of goal-creating speeds and angles shrinks. A slight overhit will go over the bar, and an underhit will bounce in front of the goal. When Zlatan hit that ball he put backspin on it to counteract air resistance, and make the path more like the gravitational parabola. As the ball span from his foot, he must have known immediately that he had done something special. The ball and curved exactly as he wanted it to, and fell perfectly into the England goal.
I watched Zlatan’s goal on TV at home with my family. My Swedish wife jumped in the air screaming with joy and performing a kung fu display. My daughter watched her mum in delight, grinning and holding her hands over her ears to cut down the noise. And my son collapsed in a new round of tears, cursing Zlatan, and sobbing that it was the substitution of his beloved Steven Gerrard that had led to the goal. This scene — in my living room, on the pitch, in Friends Arena where even the England fans applauded, and repeated across Sweden — reflects a passion for a game that is neither random nor structured.
It was, quite simply, magical.
Extract from chapter 5 of Soccermatics: Mathematical Adventures in The Beautiful Game.
