Activity: Mathematical Modelling of Free Kick

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Activity: Mathematical Modelling of Free Kick

Purpose

Mathematical modelling is a process that uses mathematics to represent, analyse, make predictions or otherwise provide insight into real-world phenomena. In order to perform this, relevant mathematical equations must be defined, assumptions must be made, and results must be produced.

In this activity you will be using a predefined Excel template to model the flight of a ball after a free kick in football (soccer). You will be guided by the PowerPoint presentation presented in the lecture and by following certain steps you will deduce the path of the ball. This activity will interest anyone with an inclination in mathematics (especially applied mathematics) In other words, you must have a strong mathematical background to be able to complete it successfully.

Activity outline

After careful study of the mathematical modelling lecture PowerPoint presentation, complete the following steps:

Open the file ball_excercise.xls located in the VLE folder Mathematical Modelling Activity. Carefully read the explanatory notes for each symbol used in the sheet. Note that certain parameters such as the initial position and velocity of the ball, the inclination of the spin axis (set at 90 degrees for pure sidespin), the elevation of the kick, etc., are already provided as inputs. Do not change these inputs!

By carefully consulting the presentation notes, perform the following:

ü Start filling in the rows for the trajectory results. In order to do this, you are reminded that:

Ø The formula for the spin parameter is provided in the notes.

Ø The drag coefficient is assumed to be constant at a value of 0.15.

Ø The Magnus coefficient is assumed equal to the spin parameter

Ø The formulas for the drag and Magnus parameters are provided in the lecture presentation.

Ø For the first row, use the initial position and velocity that are already defined in the top left corner of the sheet.

Ø For the rest of the rows, the position and the velocity can be obtained from the Euler integration method presented in the lecture PowerPoint.

Ø The acceleration can be obtained from the equations of motion provided in the lecture presentation.

Ø The formula for the total velocity is provided in the lecture presentation.

Once you are finished you should have the obtained the trajectory data for a free kick, kicked from 20 yards (18.28m) from goal, at a velocity of 25 m/s, with pure sidespin of 7 rev/s, at an initial elevation of 16 deg.

ü Plot the trajectory of the ball in two separate graphs (height vs horizontal distance from goal and lateral deviation vs horizontal distance from goal).

ü By assuming that a defensive wall is positioned 9.15 m from the free kick position and has a height of 1.8 m, check if the free kick will be successful in resulting in a goal.

ü Vary the initial elevation of the ball to 15 deg and then to 17 deg and comment on the result of the free kick and what this implies for the player taking it.

Evidence of completion

In order to provide evidence of completing this activity you must produce a report a minimum of 500 words long. There is no maximum limit set for the report. It must include some background information, details of the calculations performed, details of results produced including graphs from the Excel outputs.