Physics of bicycle kick (scissors kick) in football soccer

In soccer, the bicycle kick has provided viewers moments of breathtaking spectacle that seem virtuosic in scope. The novelty of such moments is underscored by the rarity with which players have performed this complex skill during national or international tournaments. 

The rarity of these occurrences is both a product of perceptions that it is a high-risk, low return skill and by the fact that there is a dearth of scientific research on the biomechanics of the technique. Pelé, Cristiano Ronaldo (CR7), Ronaldinho, Messi, all the best players have performed at least once a bicycle kick.
Cristiano Ronaldo best goal

This magic skill has captivated fans and also the scientific community. Some studies such as Shan et al., (2015) applied 3D motion capture and numerical simulations to better analyze and understand this movement. 

In this post we will summarize the physics processes involved in bicycle kick (or scissor kick) taking as example the famous kick from Pele, the greatest player of all times. This movement of less than 1 second will always amaze soccer fans and physics fans.

Bicycle kick movements

The movements of the bicycle kick  can be described in three phases:

1. The jumping: The player (Pelé) places himself back to the intended direction of the kicked ball and to jump, his center of gravity (CG), defined as a point where the resultant of all the weight forces can be considered acting upon, projects a little behind of his impulse foot. This allows him to gain rotational momentum when he applies force on the ground that passes within a certain distance from the CG to jump like in a back somersault. 
Barcelona Real Madrid

2. The scissors: Once completely in the air, Pelé, in astonishing synchrony with the ball trajectory, elevates the leg which is going to hit the ball and moves the other leg in the opposite direction; like the movement of a scissors, as can be seen by the angles between the thigh and the trunk for each leg shown in Figure1c. 

While the movement is performed, the head is kept in a very stable posture because he must gaze at the ball. To facilitate the rotational movement of the kicking leg at the beginning, he bends the knee of this leg, approximating his limbs to the hip, and fully extends this leg just before the kick to hit the ball as high as possible, like the divers do to rotate faster during a dive. 

The physical property being altered is called the rotational inertia, the property of a body to resist change in its state of angular movement. The rotational inertia is calculated as the product between the body mass and the squared distance from the body to the center of rotation. 

Decreasing the distances of each segment to the hip joint decreases the total rotational inertia of the lower limb. In the air, the arms stabilize the body position: the arms are kept away from the trunk in the frontal plane to intentionally increase the body’s rotational inertia in the longitudinal direction to diminish any rotational perturbation in this direction, like a circus acrobat walking on a rope with a balancing beam in the hands.

3. The ball striking: The leg, at high speed, intercepts the ball above the height of a standing person, and changes its movement. Then, for a moment (see the video), it seems as if his body stops in the air and only the legs rotate around the hip. 

This phenomenon is due to the movement of other segments of the body that move faster than the trunk: although the entire body CG is in a parabolic trajectory, as predicted by the classical laws of motion, the trunk vertical trajectory (trunk CG) actually slows down in its apex.
Pele best goal
Figure 1. Analysis of Pele's bicycle kick (Source: Demotu)

How dos GPS work? How many satelittes are required for GPS?

Several times I encountered people misunderstanding how GPS works and believes that GPS requires just 3 satellites. Other misunderstood concept regards how GPS works. Some people believe its triangulation. The real methodology is "rilateration".

In this post, we will summarize how GPS works and we will show that it requires at least 4 satellites.

How many GPS satellites are orbiting the Earth?

How GPS works?

GPS are based on trilateration, which is a method of determining the relative positions of objects using the geometry of triangles.

GPS does not use triangulation because it does not measure angles. We can visualize this concept in the following way:
  • When a GPS connects with a satellite, the GPS device estimates the distance between the device and a satellite. This distance define the radius of one sphere. Thus, the GPS device can be located on any point of the sphere’s surface.
  • When the GPS connects to a second satellite, it defines another sphere. The intersection of those 2 spheres define a circle. The GPS can be located anywhere on the surface of that circle.
  • When the GPS connects to a third satellite, it defines another sphere. The intersection of this third sphere defines 2 potential points (in a worst case scenario it may define a line segment). Thus, 3 satellites are not enough to define one single location. The GPS requires a fourth satellite to define one single location.

GPS trilateration triangulation
Image 1. GPS trilateration (Source: Trimble)

Time is the 4th variable

Some people argue that 3 satellites are enough to define one position, because one of the mentioned 2 potential locations is a not reasonable one, or non real one. This idea could be better understood by looking at the math.

When GPS connects with three  satellites, it is possible to define three distance equation with three unknown variables  (the x, y and z coordinates of the point).

The equations are quadratic ones. Therefore, sometimes one solution may involve imaginary numbers or very far location. Thus, some people  believe that 3 satellites may be enough. However, in GPS we have a fourth variable.  This 4th variable is time.

Summarizing, GPS estimates the distance to a satellite based on the travel time of a  signal emitted from the satellite.

Satellites are emitting signals at given times, and when  the GPS receives the signal it records the time when signal was received. However, the  satellite’s clocks are more accurate than the clock of the GPS device. Thus, there is a  time discrepancy that will affect the measurements and the solution. GPS solves this  problem by applying a correction factor (a 4th variable). Therefore, GPS has to solve  4 variables
  • X coordinate
  • Y coordinate
  • Z coordinate
  • C time correction factor
4 equations (4 satellite connections) are needed to solve the 4 variable.
mathematics of the GPS point distances
Image 2. Set of equation solved by GPS

Note. for simplicity I used the term GPS. However, currently the new term is GNSS (Global Navigation Satellite System)

Simulating World Cup Russia 2018

Soccer World Cup has arrived and maths are being used to analyze potential results. Spanish newspaper El Pais combined current statistics with a parametric model based on a Poisson regression and uncertainty analysis to simulate potential results and to estimate the probability of each team to win games and to become the new world champion. At the end of the post you can find the results. Much more important than the results however, is the methodology applied in the model. In this post we are interested in the maths and we present a short summary about this model. Further statistical-mathematical questions about the model may be discussed later.

The model can be described in three parts: strength of the team, simulating individual games and simulating the whole tournament.

Strength of team
Some teams (Brazil, Argentina or Germany) are stronger than others (Panama, Egypt or Saudi Arabia). This difference is quite important in this case, because national teams do not play many games together. Thus, some individual players like Neymar may be vital to win some games. The strength was calculated based on the well known Elo Rating System. This system calculates the relative skills of each participant based on its performance ratings. Although originally developed for chess players, the Elo Rating System has been successfully applied to several sports. The model from El Pais used 3 different Elo ratings. One for the players, one for the teams, and one based on the goals marked by each team.

Simulating individual games
Individual games simulate the probability of goals marked by each team. This  technique is based on a Poisson regression model proposed by Dixon and Coles (1995). Thus, the model calculates the probability of victory. The model was calibrated considering more than 17000 games. The model calibration considered difference performances for home games, away games and games in neutral field.
The calibrated model was evaluated based on the Rank Probability Score proposed by Constantinou and Fenton (2012)

Image. Calibration of the model (Source: El Pais)

Simulating the whole tournament
The previous step not only simulates the victory, but also simulates the goals. This is an important point to simulate the whole tournament. By simulating the number of goals the model can predict the first place and second place of each group; hence, defining the matches for the following stages. The last two steps were repeated 10 000 times (10 000 iterations) in order to consider different uncertainties. Although there are no details about the probability rules to define the next iteration, this was an important step because the model estimated the probability of each team to win a specific game, to pass to the next stage and to become the new Champion.

The following image shows the result of the beast teams. You can visit the whole table.
Image. Simulation result of the best teams

Satellite images from Hidroituango hydropower dam crisis Colombia 2018

Last days we have been following the Hidroituango Hydropower dam crisis in Colombia. We published posts about the crisis time frame (link to post) and the analysis from the technical committee (link to post). In this post we will not discuss about the event. Instead, we present satellite images to visualize the problem, like we did with the Oroville dam (Link to post).

Image 1 shows a comparison of the site in March 26 (before the event) and in May 17 (during the crisis). We want to point to 3 details:

  • The first noticeable detail is the water behind the dam (on the reservoir). On XXX the reservoir is empty. On the other hand, by May 17th the water level has risen so much that the reservoir is almost full. It is possible to see that the water level is very close to the spillways and the top of the dam.
  • Other important detail is the visualization of the landslides. As mentioned in a previous post, this crisis began because of landslides that blocked the tunnels. The image from May 17th clearly shows 2 landslides on the right margin of the river.
  • The third important detail is the water flow downstream the dam. The image from May 17th shows water flowing downstream the dam. This is an important detail, because the spillways were not working yet. Thus, the water flow downstream is a sign of the seepage described by the technical committee. 

Ituango Hydropower Colombia
Image 1. Hidroituango dam satellite image March 26 (Source: Planet Labs)


Flood Colombia due to hydropower dam fail
Image 2. Hidroituango dam satellite image May 17 (Source: Planet Labs)

The other image (Image 3) pair shows 2 images in an animated GIF (images from May the 02nd and May the 07th). This pair of images has less detail, but it covers a much bigger area. This second comparison shows the backwater effect of the dam. Several tributaries were flooded by the backwater effects.

flood Satellite images
Image 3. Animated GIF of images from Hidroituango (Source: GIPHY)

5 facts about the Fuego volcano eruption in Guatemala (June 2018)

Last week, Fuego volcano (Guatemala) erupted. It was the strongest eruption in several decades. We already posted some basic concepts about volcanoes. This post presents 5 facts about this eruption.

  1. The erupted ashes were about 650 degrees Celsius
  2. The erupted ashes elevated almost 10 000 m high
  3. The Fuego volcano eruption did not throw much lava. The eruption threw tons of ashes
  4. The eruption was the strongest eruption in 4 decades
  5. Currently, the main problem are the so called lahares. That is, the mixture of the volcanic ash with rain.


Check the video

Basic concepts about volcanoes and volcanic eruptions

Last week we were flooded with about volcanic eruptions in Hawaii and Guatemala. This post present some basic concepts regarding volcanoes and how volcanoes erupt?
Why do volcans erupt?

What are volcanoes?

Volcanoes are openings, or vents where lava, tephra (small rocks), and steam erupt on to the Earth's surface.

Many mountains form by folding, faulting, uplift, and erosion of the Earth's crust. Volcanic terrain, however, is built by the slow accumulation of erupted lava. The vent may be visible as a small bowl shaped depression at the summit of a cone or shield-shaped mountain.

Through a series of cracks within and beneath the volcano, the vent connects to one or more linked storage areas of molten or partially molten rock (magma). This connection to fresh magma allows the volcano to erupt over and over again in the same location. In this way, the volcano grows ever larger, until it is no longer stable. Pieces of the volcano collapse as rock falls or as landslides.

How do volcanoes erupt?

Deep within the Earth it is so hot that some rocks slowly melt and become a thick flowing substance called magma. Because it is lighter than the solid rock around it, magma rises and collects in magma chambers. Eventually, some of the magma pushes through vents and fissures in the Earth's surface. Magma that has erupted is called lava.

Magma can be erupted in a variety of ways. Sometimes molten rock simply pours from the vent as fluid lava flows. It can also shoot violently into the air as dense clouds of rock shards (tephra) and gas. Larger fragments fall back around the vent, and clouds of tephra may move down the slope of the volcano under the force of gravity.

Ash, tiny pieces of tephra the thickness of a strand of hair, may be carried by the wind only to fall to the ground many miles away. The smallest ash particles may be erupted miles into the sky and carried many times around the world by winds high in the atmosphere before they fall to the ground.

Robotic total station (demonstration videos)


A total station is an electronic surveying instrument that measures distances and angles. It can be described as an electronic theodolite with an electronic distance meter integrated with a microprocessor and a data collector storage system. Modern total stations can be divided into manual and automatic total stations.
  • Manual total stations. In manual total stations the user manually has to sight the target to the telescope, manually adjusts the horizontal and vertical screws in order to align the cross-head to the target and operates the equipment to store the data.
  • Robotic total stations. Robotic total stations perform all those steps automatically. The telescope turns automatically. Besides, automatic aiming algorithms allows the robotic total station to automatically search the target (the prism), aim at the target and follow the target.
The main benefits of robotic total stations are:
  • Faster surveying (more productivity)
  • Remote operation (less persons)
  • Avoid potential miscommunication between operator and target
There are 4 operation types of robotic total stations:
  • Automatic target aiming. By pressing a button the equipment automatically searches and aims the prism, and collects the data
  • Automatic target lock. Once the equipment defines a prism target, it locks and follows the target and collects the desired point
  • Power search. It is a combination of the previous 2. The equipment searches the target and locks to it. Then, if it loses sight of the target due an obstruction, it again searches the target and locks to it. This type is very convenient for surveying areas with interruptions due to infrastructure.
  • Imaging. The image observed by the equipment is displayed in an external screen.
The video shows a visual explanation of thefour types of operation options (link to video)