VANIER COLLEGE

The Biomechanics Behind Kicking a Soccer Ball

The Energy Demands of the Body

Raymond You

Linear Algebra II

Ivan T. Ivanov

5/19/2013

1

What is Inertia?

Inertia is defined as an object’s resistance to changes in its state of motion. In this

analysis, the text will be mainly focused on the moment of inertia. Similarly to linear inertia,

moment of inertia is the measure of an object’s resistance to angular acceleration, specifically the

resistance to rotation.

Background

The Structure of the Hip

The hip usually refers to an anatomical region or a joint. It is situated lateral to the gluteal

region and inferior to the iliac crest. The innominate bone (or hip bone) is a large flat bone,

forming the lateral half of the pelvis. It is composed of three bones fused together, usually by

adulthood. It makes up part of the hip region. The two innominate bones form the bony pelvis

contributing to the proximal articular surface of the hip provided by the acetabulum. On the sides

of the innominate bone, the ilium, ischium and pubis forms a socket to hold the femoral head of

the femur. The femur, the largest bone in the body, is divided into the head, neck and shaft.

In order to understand a kick, only the proximal end of the femur will be studied. The

femoral head is about two-thirds a sphere and is mostly congruent to the socket lateral to the

innominate bone. The congruency is meant to reduce load strain and increase contact area

between the femoral head and the socket in order to reduce the force per area (stress). The hip

joint is also known as the acetabulofemoral joint which closely resembles a ball-and-socket

model. The primary function of the hip joint is to stabilize the body in static or dynamical

situations. The hip joints are supported by the synovial capsule, attached to the bony pelvis. The

bands of fiber run parallel to the femur. The hip joint capsule (the capsular ligament) is located

proximally on the acetabulum and intertrochanteric crest. To further reinforce the capsule, three

longitudinal bundles of fibers are wrapped around the capsule: the iliofermoral, ishiofemoral and

the pubofemoral ligaments (Oatis 648).

In a simplified case, the moment of inertia of the kick will mainly have an effect on the

capsule and the ligaments of the hip. A soccer kick will cause the femoral head to rotate in the

socket effectively putting strain on the surrounding fibrous bands.

A soccer kick is described as a proximal-to-distal motion of the kicking leg i.e., the leg

moving away from the body. The motion can be broken down by segmental and joint movements

existing on multiple planes. The backswing of the thigh will initiate the kick. The thigh will

experience a deceleration due to the shanks and the hip muscles. The main muscles involved in

the forward kick are the quadriceps and knee extenders, along with the longitudinal and

circumferential fibers that hold the femoral head into the socket. The forward acceleration of the

kicking leg is provided by the knee extender as well as a motion-dependent moment from the

quadriceps (Kellis and Katis 154-155).

2

The instep form is by far the most effective method to produce a powerful kick. During

the kicking phase, the leg is drawn backwards and the hip extends by 29°. The hip rotates

externally in a forward motion following by the knee flexing. The forward motion is then led by

the rotation of the pelvis. The hip will also begin to flex as it rotates externally (Kellis and Katis

154). The analysis will however treat the kick as start and end points where the initial and final

moments of inertia are obtained through the study. Furthermore, the amount of work done will

be determined by the average force produced from the initial and final moments of inertia. The

contact between the foot and ball consumes about 15% of the energy produced from the motion

(Crosby). Nevertheless, to simplify the situation, the energy consumption will be neglected. To

perform the kick, a great amount of force is applied on the muscles responsible for the

movement. Therefore, a certain amount of energy must be supplied from the body in order to

establish this motion. This analysis will seek to determine how much energy the body needs to

provide per kick.

Fig.1.1. The femoral head in the socket is lateral to the innominate bone

supported by longitudinal bundles of fiber.

The soccer kick can be divided into 6 frames, where the body is posed in certain

positions. For this analysis, frames 3 and 6 are of interest. Position 3 is the position right before

the forward motion of the legs. Position 1 and 2 are the approach and wind up of the leg for the

full backswing. Frame 6 is the extension of the knee, approximately parallel to the horizontal

line. It is the follow-through of the kick.

3

Fig.1.2. The six positions of a soccer kick, where the main position of interest in

this analysis is frame 3 and 6. Frame 3 is the final position of the backwards wind up, and frame 6 is the final position of the

forward swing.

It should be noted that this is the coordinate system that will be followed.

Fig.1.3. The coordinate axis system throughout the investigation

Analysis of the Inertia of a Soccer Kick from Initial Position

To begin the analysis, the first frame of the motion will be as depicted a simplified image

demonstrating position 3 of Fig1.2.

Fig.1.4. The initial position according to the analysis. This is the transition

between the backswing of the leg to a forward swing; swing-limb loading.

The moment of inertia of a body is calculated by summing the product of mass and

distance squared for every particle in the body with respect to the axis of rotation.

4

The moment of inertia is given by

,

(1-1)

where m and r2 are the mass and distance squared with respect to the axis of rotation (the center

of mass of the body).

Moment of inertia is also dependent on the magnitude and distance of each point masses

relative to the axis of rotation. The further away particles are from the axis of rotation, the

greater contribution it’ll have to the overall moment of inertia. In the case of a rigid body, the

further away any body segment is from the center of rotation, the greater the segment will

contribute to the moment of inertia with respect to the center of mass.

In this analysis, inertia matrices will be used with the individual unit vectors to determine

the inertia dyadic. The different body segments are rotated with respect to the hip. The parallel

axis theorem is then used to determine the moment of inertia of the rigid body in reference to the

chosen axis of rotation.

The parallel axis theorem for moment of inertia about a new axis of rotation is given by

,

(1-2)

where Icm is the moment of inertia at the center of mass of the body segment in study, m is the

mass of segment and r is the perpendicular distance from the axis of rotation.

Information for Icm is obtained from “Principles of Biomechanics” where the moment of

inertia of the segment in study has already been determined. The study will focus on determining

the second moment of inertia; mr2.

The inertia dyadic is determined relative from point P (the segment of the body) relative

to the origin O (the hip) for the direction of the unit vectors, given by

( ),

(1-3)

where ux,y,z are the individual unit vectors parallel to the x, y and z axes, rPG is the position

vector relating the position of P to the origin O and M is the mass of the body segment. IP/O

is the

second moment of inertia.

Notice that this is the equivalent to Eq.1-1 for moment of inertia, except now it is

expressed as a matrix.

5

PG is the position vector relative from the origin O (hip) to the center of mass of the

segment in study, following an XYZ coordinate system. The leg is assumed to rotate about the Z-

axis; the effective rotation is on the X and Y plane, where the leg follows an angle of 29° (Kellis

and Katis 154). The PG vector presents the position in its X, Y, and Z components.

PG=

The parallel axis theorem states that the moment of inertia of a rigid body can be

determined in reference to the body’s center of mass. Following Eq.1-3, the perpendicular

distance r multiplied by the vector PG will result in the vector that can be used in order to

determine the position of the body segment in 3-dimensional space with the relative distance

from the origin O to the center of mass of the segment; rPG is the distance vector.

=

Following this, the distances in Ux, Uy and Uz can be determined by breaking down the

direction vectors and taking the cross products with the vector rPG.

ux=

uy=

uz=

6

( )

The vector rPG appears twice which accounts for the r2 in Eq.1-1 and Eq. 1-2.

The parallel axis theorem will correct the moment of inertia that is distal from the center

of mass. The moment of inertia at the center of mass of the body must be added to the second

moment of inertia produced by the segment a distance away from the axis of rotation as indicated

by Eq.1-2.

ITotal= Icenter of mass+mr2

= +

=

Notice that the resulting moment of inertia matrix is positive and symmetrical. Real and

positive eigenvalues are expected. Through an eigenvalue analysis, the values were

λ1= 0.0705999999993217

λ2=0.123391812833638

λ3=0.0707918128329600.

The positive and real eigenvalues verifies the process, and the moment of inertia can be

interpreted with real physical sense.

A similar procedure is done for the lower leg and the foot. For the lower leg, the distance

from the center of mass is the entirety of the upper leg and half the lower leg. The foot will

subsequently follow, where the distance is the entire upper and lower leg.

7

For the lower leg, the added distance of the upper leg will be incorporated in the distance

vector.

(

)

(

)

The unit vector in the y-direction implies that the rotation of the lower leg only occurs in

that specific axis.

The procedure to determine the second moment of inertia is the same for the lower leg as

for the upper leg, only differing in the new rPG vector.

The resultant inertia matrix is:

=

=

The moment of inertia matrix is positive and symmetrical and similarly, positive and real

eigenvalues are expected. Following an eigenvalue analysis, the following results were yielded.

λ1= 0.0502694938574550

λ2= 0.00589999999654500

λ3= 0.0451694938540000

The positive and real eigenvalues verifies the process.

To determine the next distance vector for the foot referring to the center of mass, both the

length of the upper and lower legs must be added together.

8

=

The results for the foot are as follows:

=

=

The eigenvalue analysis resulted in positive and real values.

λ1= 0.0715613286919120

λ2= 0.00348931427908802

λ3= 0.0750506429710000

Again, the process to determine the second moment of inertia for the foot is exactly the

same for the upper and lower leg, only differing in the distance vector rPG.

The total moment of inertia produced from the initial position would be the sum for all

the moment of inertia produced by each body part distal from the center of rotation.

9

=

Eigenvalues:

λ1= 0.236310447699954

λ2= 0.0867926163750063

λ3= 0.188903064074960

The eigenvalues were real and positive, verifying the procedure.

Next is to determine the rotation about the axis. The leg is assumed to mostly rotate in the

z-axis with a slight rotation in the y-axis as the motion of the kick is not completely rigid.

Assume that rotation in the y-axis is about 10°.

The vector defining this rotation of the leg will be given by,

V=

And therefore the rotation about the axis is determined by

10

Iv, v=

= 0.1612

.

Analysis of the Inertia of a Soccer Kick from Final Position

Fig.1.5. The final position according to this analysis; this is known as the follow-

through phase of the kick.

Similar calculations are done with the final position of the kick. However, the PG position

vector changes. This is due to the kicking leg being parallel to the x-axis. The position vector

becomes:

PG=

By changing the position of the kicking leg, the moment of inertia produced by the upper

and lower leg and the foot will be equal to the sum of each individual inertial in relation to the

distance from the origin.

After similar calculations, the moment of inertia of the final position was determined.

ITotal for Final Position=

11

Positive and real eigenvalues verifies the procedure.

λ1= 0.0773000000000000

λ2= 0.282251050000000

λ3= 0.225351050000000

To Determine the Moment of Inertia for the Rotation about the Axis

The same procedure is used for the final moment of inertia as for the initial moment of inertia.

Iv,v=

= 0.2801

Following this, converting

to

to SI units will facilitate the conversion of moment

inertia into energy.

Since 1

is equivalent to 157.087464

then the initial position will have,

=

=25.33

And the final position will have,

=

=44.00

.

Once these results are obtained, conversion to torque will require the angular acceleration.

12

Angular acceleration is defined as the rate of change of angular velocity. The SI unit

for angular acceleration is

.

To determine the angular acceleration, the rotation of the hips must be determined in

radians. To do that, the distance it travels in radians must be determined. Due to the motion of

the kick being a rotation, the toe-to -toe distance travelled from the initial to final position will

give an indication of the angular acceleration.

Fig.1.6. The black figure is the initial

position of the kick, whereas the red figure is the final position of the kicking leg. The rotation was nearly 180°. From empirical

observations, the angle from the toe in the initial position is approximately 20° below the horizontal axis. The toe-to-toe distance

travelled is determined to approximately be 3.17789 m.

The radius of the circle will be approximately the length of the leg; 1.138m. With the

known angle of the circular path, the arc length can be determined. This will be important in

finding the energy done by the motion.

This information will be valuable later.

13

To establish the angular acceleration of this kick, the duration is also need. Assume a

kick on average lasts for 0.66 seconds.

Determining Angular Acceleration

= 6.41

Since torque is

,

then the torque produced from the initial frame is

) (6.41

)

=162.40

and that produced for the final position is

) (6.41

)

=282.08 .

It is assumed that the longitudinal fibers in the hips are about 15 cm long. The force can

be determined on the muscles responsible for the action.

Initial Force

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Final Force

To find the average force, half the sum of the initial and final forces will give the force

required to rotate the motion of the kick from the initial to final position.

To determine the energy required, work must be ascertain. Work is defined as the product

of force by distance.

Work Done by the Motion

Concluding Remarks

In order to determine how much energy the body requires to perform this motion, a

comparison is made with food calories to the energy in joules needed for the kick. On nutritional

information panels, the energy provided by the food is given in calories. This is misleading as the

units are actually kilocalories. The word ‘calorie’ is capitalized in some cases to denote 1 kcal

(Layton). For instance, a jellybean usually has 4.1 Calories (caloriecount.com), but in reality that

is 4.1 kcal or 4100 calories. In SI units, a calorie is equivalent to 4.148 Joules.

If the kick produced of energy, the conversion to kilocalories would be

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Then, 1.125 kcal in terms of the label on the food products would approximately be 1

calorie.

The body will convert food to metabolic fuel via various energy pathways. The main

source of energy used by the body is ATP (adenosine triphosphate). ATP is usually converted

from macromolecules such as carbohydrates, proteins and fat obtained through eating; however

carbohydrates will generally be the main source of fuel. In order to synthesize ATP, the

macromolecules will undergo an aerobic (using oxygen) or anaerobic (without oxygen)

respiration. There are two sub-categories of the anaerobic pathway known as the ATP-CP

anaerobic energy pathway and fermentation. The ATP-CP energy pathway supplies energy for

about 10 seconds by using the ATP stored in muscles (gives 2-3 seconds worth of energy) and

CP (creatine phosphate; gives 6- 8 seconds of energy) to make more ATP. It is mainly used for

explosive and quick motions. It does not require oxygen. Once ATP from the muscles and CP are

depleted, the body will either move to fermentation or aerobic respiration. Fermentation enables

the muscle cells to produce small amounts of ATP through glycolysis in the absence of oxygen

for short and high intensity burst of physical activity. In fermentation, pyruvate (a product of

glycolysis) will be reduced to lactic acid. The partial breakdown of glucose by fermentation will

only be sustainable until the lactic acid builds up significantly in the muscle, known as the lactate

threshold. Lactic acid accumulation causes muscle fatigue, pain and a burning sensation

noticeable within several minutes. Aerobic respiration will mainly fuel long lasting activities,

and use oxygen to make ATP as opposed to the other two forms of energy pathways (Quinn).

Kicking is mainly an anaerobic procedure, notably dependent on the ATP-CP energy

pathway (Crosby). The motion is very short lived and is usual a sudden movement requiring

short bouts of energy to follow through the motion. Considering that a soccer player may be

running and performing other physical activity for extensive period of time, the body will require

a steady source of energy. It will draw the energy required from the aerobic system as opposed to

anaerobic. If a jelly bean were to be consumed, the useful nutrient such as glucose could undergo

either aerobic or anaerobic respiration to produce a usable form of energy depending on the

situation. Glycolysis is the first step in extracting the energy from the jellybean. The glucose

undergoing glycolysis will produce pyruvate which can either lead to anaerobic respiration (in

the absence of energy) or will follow with the Citric Acid cycle (in the presence of oxygen).

Which pathway the pyruvate undergoes is largely dependent on the condition of the body.

In principle, one jellybean should provide enough energy to go through with at least 3

soccer kicks if only the energy provided is considered. In reality, the nutrients received from the

jellybean may be treated in several of ways and will ultimately be useful to the body one way or

another.

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Work Cited

Crosby, J.. N.p.. Web. 19 May 2013.

<http://www.sportsinjurybulletin.com/archive/biomechanics-soccer.htm>.

"Calories in Candies, Jellybeans." Calorie Count. N.p., n.d. Web. 19 May 2013.

Huston, Ronald L. Principles of Biomechanics. Boca Raton: CRC, 2009. Print.

Kellis, Eleftherios, and Katis, Athanasios. "Biomechanical Characteristics and

Determinants of Instep Soccer Kick." Journal of Sports Science and Medicine 6 (2007): 154-65.

Print.

Layton, Julia. N.p.. Web. 19 May 2013. <http://science.howstuffworks.com/life/human-

biology/calorie.htm>.

"Moment of Inertia." Moment of Inertia. N.p., n.d. Web. 11 May 2013.

Oatis, Carol A. Kinesiology: The Mechanics and Pathomechanics of Human Movement.

2nd ed. Philadelphia: Lippincott Williams & Wilkins, 2009. Print.

Quinn, E.. N.p.. Web. 1 Jun 2013.

<http://sportsmedicine.about.com/od/sportsnutrition/a/Energy_Pathways.htm>.