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Aerodynamic Forces of a Soccer Ball

Please note: the simulation below is best viewed on a desktop computer. It may take a few minutes for the simulation to load.

Many students enjoy playing and watching the game of soccer (which is known as “football” in every country in the world except the United States). With this simulation you can investigate how a kicked soccer ball “bends” as it flies through the air. A soccer ball encounters aerodynamic forces as it flies and you can vary the values of the factors that affect the size of the forces on the ball. These are the same forces that generate the lift and drag of an airplane wing and produce a big league curve ball.


The flight of a soccer ball is a fairly complex physics problem. The flight path is described by integrating Newton’s second law of motion. The external forces on the ball are the weight, the aerodynamic side force, and the aerodynamic drag. The weight is constant, but the magnitude of the aerodynamic forces depend on the air density, which is a function of location and local weather conditions, and the square of the velocity, which changes during the flight. The resulting flight path is three dimensional, so you must specify several parameters to define the initial conditions when the ball is kicked. This web page describes the variables required for the calculation and how to set the values for the variables using the graphical user interface for the computer program.

General Instructions

This program is designed to be interactive, so you have to work with the program. There are a variety of input “widgets” that are used in the graphical user interface. There are several colored buttons used by the program. To operate a button, use the mouse to move the cursor over the button, and click the left mouse button. At any time, to return to the original default conditions, click the orange Reset button at the upper right of the program. A choice box has a descriptive word displayed in a box with an arrow to the right of the box. To make a choice, click on the arrow to display the choices, move the cursor down the selection menu and click to make your selection. Some variable choices are made using round radio buttons. To make a choice, move the cursor onto the round button and left click. A small black dot appears in the chosen radio button. Current values of the design variables are presented to you in text boxes. To change the value in an input box, select the box by moving the cursor into the box and clicking the mouse, then backspace over the old number, enter a new number, then hit the Enter key on your keyboard. You must hit Enter to send the new value to the program. For most input variables you can also use a slider located next to the input box. Click on the slider bar, hold down and drag the slider bar to change values, or you can click on the arrows at either end of the slider.

If you experience difficulties when using the sliders to change variables, simply click away from the slider and then back to it. If the arrows on the end of the sliders disappear, click in the areas where the left and right arrow images should appear, and they should reappear. These are known Java problems.

Screen Layout

The program screen is divided into two main parts:

  1. On the left side of the screen is the graphics window. The graphics window has four possible displays: a downfield view, top view, side view, and TV view. The TV view is a perspective view from the sidelines. You choose your view by clicking on the appropriate box at the top of the graphic and the current choice is by a yellow “light”. You can move the graphics side to side and up and down by moving your cursor into the window, holding down the left button on the mouse and dragging the graphic. You can “zoom” into the graphic by using the slider at the bottom of the window. Click on the yellow vertical bar, hold down and drag the bar from side to side. If you lose the picture, click on the “Find” button at the upper right to restore the default graphics location and size. Above the graphics window is a picture of a spinning soccer ball. The spin axis of the ball is shown on the ball and you can adjust the axis using the input devices described below. By adjusting the spin axis, you can make the aerodynamic force cause the ball to bend, lift, or dive. If the spin axis is vertical, the ball curves from side to side, which soccer players call a “bend”. If the spin axis is horizontal, the ball lifts or dives depending on the direction of the spin. If the axis is between vertical and horizontal, you get a little bit of both effects. You can see the effects of spin and gravity on the vertical flight path by choosing the side view. You can see the side to side effect on the horizontal side path by using the top view. You can see the combined effects of aerodynamics and weight in the downfield view and the TV view. We have drawn the goal and net in each of the view windows so that you can adjust the input parameters to score a goal. For the corner kick, the objective is not to score a goal, but to put the ball in front of the goal with sufficient height so that a teammate can “head” the ball into the goal. So for the corner kick, a blue target box will appear in front of the goal and your job is to get the ball into the target box.
  2. The input parameters for the kick are specified on the right side of the screen. Beginning at the top, we have four buttons that initiate the kick and control the graphics. The green “KICK” button begins the solution of Newton’s equation to determine the trajectory. When you click on the “KICK” button, the boxes below the buttons display the time, speed, and magnitude of the forces during the flight of the ball. The blue “Save” button can be used to freeze the graphics of a completed kick for comparison with a later kicks. The white “Clear” button erases all the graphics and prepares you for the next kick. The orange “Reset” button can be used at any time to return the program to its initial conditions.

Immediately below the colored buttons are some radio buttons to choose the type of kick. For a penalty kick the ball is placed in the middle of the field, 36 feet from the goal line. You cannot change the location of the ball for a penalty kick. You can change the angle at which the ball is kicked using the boxes and sliders at the lower right. The vertical angle is the angle formed between the direction of the kick and the ground, while the side angle is measured from the downstream (towards the goal line) direction. For a free kick, the ball is initially placed 60 feet from the goal line and 30 feet from the centerline of the goal. For a free kick, you can change the location using the input boxes or sliders located at the bottom right. Y – side is measured from the centerline of the goal, and X – distance is measured from the goal line. For a corner kick the ball is placed in the corner formed by the sideline and the goal.

You can kick with either your right foot or your left foot by clicking on the round radio buttons located below the type selection. The direction of rotation is reversed between right foot and left foot. The Spin Axis angle (degrees) and the Spin -rpm (revolutions per minute) are set by using the input boxes and sliders at the lower right listed under Kick Input. As mentioned above, if the spin axis is vertical (angle equals zero), the side force is purely horizontal and the the ball moves left or right depending on the direction of rotation. If the axis is horizontal (angle equal to +/- 90 degrees), the side force is vertical and lifts or dives the ball, depending on the direction of rotation (+/- value of spin). The amount of the force and flight trajectory also depend on the initial speed of the ball. You set the speed using the box and slider in the Kick Input. Finally, the value of the side force depends on an efficiency factor called the lift coefficient Cl. The default value for a spinning soccer ball is .25. You can change the value of Cl as described below. Besides the side force, there is an aerodynamic drag generated on the ball which opposes the motion. The size of the force depends on the square of the speed, the density of the air and the value of the drag coefficient – Cd. The default value of the drag coefficient for a soccer ball is about .25. To allow you to study the effects of lift and drag on the flight of the ball, you can select to input a lift and drag coefficient by using the radio buttons at the upper right. Select “Input” and then enter in the value in the input boxes labeled Cd and Cl.

Below the flight data boxes are some choice buttons to select the stadium location and the weather conditions. Changing the stadium location changes the atmospheric conditions because the atmospheric conditions depend on altitude. We have included all of the stadium locations used during the 2010 World Cup so that you can study how altitude effects the flight of the ball. The default weather conditions are an Average Day at the selected stadium. The average is based on an NASA model of the atmosphere and how the pressure and temperature change with altitude. For the average day, we have the temperature set to 59 degrees at each stadium. A Hot Day sets the temperature to 89 degrees, and a Cold Day sets the temperature to 35 degrees. The Hot, Humid Day includes the effects of relative humidity. The program calculates the air density that corresponds to the temperature and pressure through the equation of state.

The objective of the corner kick is to put the ball into the target box. The objective of the penalty kick and free kick is to put the ball into the goal. Try setting up a kick that scores a goal and then change the stadium location. What happens to the trajectory? (In soccer, scoring is much harder than it looks!)

Have fun!

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