Balance Ball: Introduction

Balance Ball	VRML Project

Mathematics/Physics

Our goal is to describe the whole model with physics to achieve a simulation that comes as close as possible to the reality. Here are some basic thoughts: For simplification, we describe the model with only two instead of five balls.

The main rule of physics we have to apply here is the law of impulse, meaning the law that describes what happens if two bodies with mass m collide. This is dependent upon the velocity of both masses, expressed with v₁ and v₂.
In a first step, we are interested in v₁ expressed with earth gravity g, mass m and height of the ball h. l is the length of the string where the balls are attached. Using the law of energy, we get:

Using the formulas above, we obtain the following for the velocity v₁:

or, expressed by angle a:

Now that we have the velocity, we can go on with our calculations.
There are three different cases we use to describe the collision of two bodies. We have

Fully elastic collision
Absolutely no deformation of the two bodies occurs. That means that we do not have an energy loss for the deformation of the bodies.

Partially elastic collision
A part of the energy of motion is lost in deformation when the two bodies collide.

Non elastic collision
If the two bodies collide, we have the effect that both bodies stick to each other. Therefore they will have the same velocity v after the collision.

To describe this behavior, we introduce the factor k.

fully elastic: k = 1
partially elastic: 0 < k < 1
non elastic: k = 0

Looking in formula tables, we found the following formulas:

and

v’₁ and v’₂ are the velocities after the collision. Since in our model the masses m₁ and m₂ are identical, these two formulas can be simplified as follow:

and

These formulas have the disadvantage that they do only describe the impulse on one axis, in our example in x direction. Thinking that our system is partially elastic, we can conclude that collisions will happen in different positions of the balls, meaning that the collision direction is also different. Therefore we will have to introduce vectors for the velocities of the balls.

Another problem that has to be solved is the calculation of the different collisions. So far we figured out that VRML does not support a collision detection between two objects, therefore we will have to implement it. These will bring other calculations like determining the positions of the different balls.

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