Group Simulation Model Analysis
Bouncing Ball Model: Use of Zero-Crossing Detection
2012

Executive Summary
In the system of a bouncing ball there are many factors which influence how the ball bounces. The 5 main components which affect how high the ball will bounce include the initial position of the ball, the initial velocity of the ball, the elasticity of the ball, the gravity of the selected area, and also the temperature of the material in which the ball was made from. These 5 major components of the basic system determine how high the ball bounces and how much energy was lost in the process of bouncing as well as allowing us to determine the relationship between these components and how each of these affects the overall system as a whole. However by changing these components of the system, it will dramatically change the overall results.

In the system of a bouncing ball there already at secondary school level, illustrates Newton’s laws of motion and concepts of gravitational energy and kinetic energy with examples of objects dropped or thrown vertically and contains investigative activities about falling objects, the physics and mathematics. The fives main components which affect how high the ball will bounce they are initial position, the initial velocity, the elasticity, the gravity and the temperature of the material in which the ball was made from. These five major components of the basic system determine how high the ball bounces and how much energy was lost in the process of bouncing as well as allowing us to determine the relationship between these components and how each of these affects the overall system as a whole. However by changing these components of the system, it will dramatically change the overall results.

Table of Contents Executive Summary i 1 Introduction 3 2 System Overview and System Behaviour 4 Velocity 4 Height 4 Ball Elasticity 5 Gravity 6 3 Sensitivity Analysis on Different Types of Parameters 7 Initial Velocity 7 Initial Position 10 Elasticity 12 Gravity 13 Case 1 13 Case 2 14 Temperature 15 4 System Refinements 16 5 Conclusion 17 6 References 18 7 Appendices 19

1 Introduction
By many, bouncing a ball is seen to be a simple task; however there are several factors which affect the systems as a whole. This report aim is to identify the major components of this particular system and examine how changes one of these components will affect the way the ball bounces. This report also aims to explain some of the key variables involved when bouncing a ball, such as the elasticity of the material, the height dropped, the initial velocity of the ball, the gravity in which the ball is dropped, the mass of the ball, the hardness of the surface and also the temperature of the materials involved. All of these variables have a major impact on how the ball will bounce, as well as change the overall behaviour of the bouncing ball itself. This change will be determined through a sensitivity analysis of the most influential parameters of bouncing ball system such as the initial velocity and height. These will be further explained through the comparisons of various components and how they change depending on the other variables which helps to determine the relationships between the components in the system to see how they combine together to create a system which seems simple to a normal person, but in reality involves several components not visible in order to function as it does.
A bouncing ball model is a classic example of a hybrid dynamic system. A hybrid dynamic system is a system that involves both continuous dynamics, as well as, discrete transitions where the system dynamics can change and the state values can jump. This report aims to identify the major components and explain how changes the way the ball bounces will affect by the components, and to examine the key variables involved when bouncing a ball: the elasticity of the material, dropped by the height, the initial velocity, the gravity, the mass, the hardness of the surface and the temperature of the materials of the ball. These variables will affect on how the ball will bounce, and they can change the overall behaviour of the bouncing of the ball. The initial velocity and height are main of change will be determined through a sensitivity analysis of the most influential parameters of bouncing ball system. In this report will be further explained through the comparisons of various components and how they change depending on the other variables which help to determine the relationships between the components in the system.
References
1. Bernd Ulmann, Simulating a bouncing ball in a box, http://www.vaxman.de/analog computing/bouncing ball bouncing ball.html
2. S. Ulam On some statistical properties of dynamical systems J. Neyman, J.C. Oxtoby (Eds.), Proceedings of the fourth berkeley symposium on mathematical statistics and probability, vol. 3University of California Press, Berkeley (1961), pp. 315–320

2 System Overview and System Behaviour
The diagram below shows the factors that affect the bounce of a ball:

Figure1: Diagram of Variables affecting Bounce of the wall
Ball Material (Elasticity)
Elasticity is the ability inherent in most solid material that allows the deformed body of the material to return to its original shape and size when the forces causing the deformation are removed. In this simulation, it is the elasticity of the rubber ball that enables it to bounce for a period of time before it loses its kinetic energy through the continued bounce. The kinetic energy of the ball will reduce each time the ball bounces of a surface, converting kinetic energy into thermal energy due to the friction of the molecules within the rubber ball rubbing against each other when the ball is deformed as it hits the surface of the floor. What this means is that the ball would continue to bounce until it loses all its kinetic energy and finally would come to rest. [4]
In the context of this simulation, we can compare elasticity and how it affects different materials for example between a rubber ball which has high elasticity would bounce of a surface when dropped from a height, but a ball made from plasticine would not bounce at all it has very low elasticity and it won’t retain its shape after the deformation when dropped.
Height
The Initial Position also known as the height is one of the major components in the model. At the Initial height (assuming the ball is being dropped from a stationary position), the ball has Maximum potential energy. Once the ball is released some of the potential energy changes into kinetic energy which increases as the ball gets closer to the ground and is at a maximum right before impact, during which the potential energy is almost zero. Gravity then pulls the ball downwards and using that kinetic energy to drag the ball down towards earth. When a rubber ball hits the floor, kinetic energy spreads out over the floor, and then transfers back into the rubber as it returns to its original shape. It does this so fast, it actually pushes against the ground and shoots the ball into the air.
The velocity at which the ball will bounce is directly proportional to the initial height at which the ball is dropped. At different heights of which the ball is dropped, the time taken for it to reach the ground will be different due to the different amount of kinetic energy. [2]

Figure 2: This drawing shows the transfers of energy that occurs when a bouncy ball is dropped onto a hard surface. [3]

Velocity
If a ball is dropped or is free falling, the initial velocity is zero. This means the ball travels at a rate of 0m/s and then its velocity will be is the same as its acceleration due to gravity. On the other hand, if the initial velocity of the ball is different from zero that means greater or less than zero, (depending on which direction is taken as positive by the observer) then the balls velocity follows the gravitational acceleration of that particular geographical place. In this case, the ball's velocity will still follow its acceleration due to gravity, but will just start at a faster rate.However, the velocity will decrease as it bounces until it stops due to free falling body.[1]
Gravity
One of the most influential parameters is the acceleration due to gravity. Gravity plays a major role as a natural force directed towards Earth’s centre due to Earth’s rotation. When a rubber ball is thrown upward, it will naturally fall back to the ground. This is due to the gravitational force that pulls every object towards Earth’s centre. Gravity on earth is a constant and is approximately 9.8 m/s2. In other words, the inward pull towards Earth'scentre, or gravity, causes a ball to accelerate at 9.8 m/s2 in a direction pointing towards the earth. Since gravity is a constant, balls of different masses will fall at the same rate: 9.8 m/s2.[1]

The Earth's one natural satellite, the Moon, is more than one quarter the size of Earth itself (3,474 km diameter) and has the gravity one-sixth of the Earth's gravity. Acceleration Due to gravity on the Moon is said to be 1.6m/s2. [5]

3 Sensitivity Analysis on Different Types of Parameters
Initial Velocity
As you can see in the figures below however the bouncing ball in this particular scenario has an initial velocity of 15m/s, 20m/s and 50m/s. As shown in Figure 3, the amplitude is slightly greater than 20 and it time offset is approximately 20. Figure 4 shows that as the initial velocity increase by a quarter of the first initial velocity the amplitude reaches 30 and the time offset increased too by 4. In similar fashion, by the third trial, by increasing the first initial velocity in 2.5 folds, the amplitude and the time offset is increased by approximately by 6.5 folds and 2.5 folds respectively. This illustrates that as the initial velocity of the ball incremented the amplitude and the time offset of the bouncing ball increase simultaneously.

Figure 3: Simulink Model with 15 m/s velocity

Figure 4: Velocity and Displacement plot with 15 m/s velocity

Figure 5: Simulink Model with 15m/s velocity

Figure 6: Velocity and Displacement plot with 20 m/s velocity

Figure 7: Simulink Model with 50 m/s velocity

Figure 8: Velocity and Displacement plot with 50 m/s velocity

Initial Position

Figure 9: Simulink Model with 10 m initial position

Figure 10: Velocity and Displacement plot with 10 m initial position
The height at which the rubber ball was dropped from initially was 10m as shown in Figure 2 and 3. At the 10m drop the graph showed a first bounce after 3.62 seconds which reached 0m(the ground) and bounced to a height of 13.74m at 5.3 seconds. It continued bouncing for about 12-13 times before coming to a stop.
Figure 11: Simulink Model with 50 m initial position

Figure 12: Velocity and Displacement plot with 50 m initial position
When I changed the height from 10m to 30m as shown if Figure 4 and 5, the graphshowed the first bounce at 4.44 seconds which reached 0m and bounced to a height of 26.5m at 6.8 seconds. It continued bouncing for about 8-9 times before coming to a stop.This shows that if the ball is dropped from a higher height it will have a higher bounce after coming into contact with the ground and the velocity will increase.
Elasticity

The main factor allowing for the bounce of a ball is its elasticity. Its elastic properties reduce the amount of deformity that occurs when it impacts on a surface, making the ball more effective at bouncing. So, if a typical rubber ball has an elasticity of 0.8, it will bounce much higher and for a longer period of time, when compared to a ball with a lower elasticity, for example,

Figure 12: Velocity and Displacement plot with elasticity of 0.8, 0.65 & 0.45

The following figure shows that with an elasticity of 0.65 the ball only manages to bounce for a time offset of about 12. This is in contrast to the longer time offset seen in this pervious simulation which tested the rubber ball at an elasticity of 0.8. The change in velocity and displacement can be observed through the 3 figures, showing how elasticity affects these variables. The rubber balls with lower elasticity would have a significantly lower displacement from the surface and loses much more energy during this process. Thus, due to the much larger energy loss due to the lower elasticity’s, figure 2 and 3’s rubber balls can only bounce for a short period of time.

Gravity
Case 1
Figure 1 below is the Simulink Model of the bouncing ball system, with an initial velocity of 15m/s and the position of the ball as a function of time is 10m above the ground, and gravity = 9.81 m/s2which is the acceleration due to gravity.

Figure 13: Simulink Model with gravity of 9.81 m/s2

Figure 14: Velocity and Displacement plot with gravity of 9.81 m/s2

* The initial velocity of the ball = 15m/s * The ball is thrown upward until reaches its maximum height of 22m and falling with an acceleration of 9.81m/s2. The ball took 4s for the first hit on the ground after reaches the maximum height continues rebound after hitting the ground until the bouncing eventually dies out after 20s.
Case 2
In this case, the initial velocity and position of the ball are set unchanged with the first modelling. Instead of using 9.81m/s2, assume that the ball is thrown on the Moon, that having 1.64m/s2 acceleration due to gravity of the Moon. (Gravity on the moon is one-sixth of gravity on Earth)

Figure 15: Simulink Model with gravity of 1.635 m/s2

Figure 16: Velocity and Displacement plot with gravity of 1.635 m/s2, Stop time = 20& 50

Figure 17: Velocity and Displacement plot with gravity of 1.635 m/s2, Stop time = 100

Temperature
As a ball bounces, it traps heat and warms up. Energy is always being renewed and transformed when the ball is bouncing. Rubber balls with firmly packed molecules lose little energy to heat and surface deformation and will bounce better under different temperatures. Basketballs and soccer balls perform better when the temperature is hotter because the air molecules in the ball will expand thus overinflating the ball and it won’t easily lose its shape on contact. Where else on colder days, the air molecules will contract and the molecules in the ball will under deflate and have less elasticity.The material inside golf balls, baseballs and softballs with rock-solid cores becomes more or less elastic, depending on temperature under this same expansion/contraction principle.[1]

4 System Refinements
The simulation model short to include how is the bouncing ball affected by the surface on which it bounces. [6]
For example, if the surface is spongy and sandy the amount of energy loss is much higher than dense and smooth surface. This means, the ball eventually stops within a short period of time. Additionally, air friction acting on the ball and the angle of which the ball may throw is neglected. Assume that the ball is thrown an angle of 45˚ horizontally, the magnitude must be multiplied by cosine 45° of the position and velocity vector; and also, air friction influences the ball movement. The other flow of the system is making the gravity constant. It is better to make this component is a variable, because the gravitational force of the earth is different in different geographical location. Furthermore, acceleration due to gravity is different in different planet. For instance, the gravitational acceleration on moon is one sixth of the gravitational acceleration on earth. [7]
Therefore, adding the bouncing surface energy absorption rate, the influence of air friction towards the ball and the angle the ball strikes makes the system more dynamic as well as change the gravity from constant to variable. [7]

5 Conclusion
In a simple system such as bouncing a ball, there are many components and variables which largely affect how the ball bounces. Because of this it is essential to take into account all components of the system in order to correctly identify the major influences. For this particular system, the 5 major variables determined through the testing of how each variable influences the system, it was found that the initial velocity of the ball, the initial position or height of the ball, the gravity of the planet, the elasticity of the material, and the temperature of the material have the greatest impacts of the system as a whole; however every component affects the system in some way, and therefore all components and variables must be analysed to a great extent when determining how any single variable or component changes the effectiveness of the system.

6 References
[1] B. Smith, et al. (2011, 14.10.2011). Bouncing Balls. Available: http://www.unc.edu/~brads/bouncingballs.html

[2] M. Osborne. (2011, 24.09.2011). Why Do Balls Bounce Differently? Available: http://www.livestrong.com/article/147292-why-do-balls-bounce-differently/

[3] ScienceBuddies. (2011, 12.10.2011). On the Rebound: THe Height Limits and Linearity of Bouncy Balls. Available: http://www.sciencebuddies.org/science-fair-projects/project_ideas/Phys_p071.shtml

[4] A. Z. Jones. (2011, 14.10.2011). Major Laws of Physics. Available: http://physics.about.com/od/physics101thebasics/p/PhysicsLaws.htm

[5] R. Russell. (2005, 09.10.2011). The Earth's Moon. Available: http://www.windows2universe.org/earth/moons_and_rings.html

[6] (2003, 24.09.2011). The Energy of a Bouncing Ball. Available: http://galileo.phys.virginia.edu/outreach/8thGradeSOL/EnergyBallFrm.htm

[7] I. Usal. (2011, 14/10/2011). Gravitation: Physics. Available: http://www.kirupa.com/developer/actionscript/gravity.htm

7 Appendices Section | List of contribution | Cover page | Dinesh | Executive summary | Kristoffer | Introduction | Dinesh, Kristoffer | System overview & System Behaviour | All members | Sensitivity analysis | All members | System Refinements | Tesfaye, Steven | Conclusion | Mohd. Hariz | References | Steven | Appendices | Mohd. Hariz , Tesfaye | Signatures of All Group Members |