modelling : physics I

# introduction

A simple model uses physics and mathematics to compute positions, velocities, and accelerations of all objects in the scene.
But for heavy objects accelerations may not be useful. Think of a person who wants to move a house. If this person is not
a hero the house will rest at its position - no velocity, no acceleration, although the person applies a force on the house.
Thus non moveable objects can be left out when computing and applying forces.

A practial method for integration complex formulae or solve differential equations is **iteration** or **finite integration**.
Instead of trying to solve integration schemes which may not lead to a solution, finite steps are applied instead of infinitesimal ones.
But the increment is important whether you get a good result or nothing but pain.
During iteration the Nyquist criterion will play an important role as you will see, but I will go into detail in the subsections.

In all equations the following typesettings are used:

**bold** | for vectors **x** in R^{n}, |

*italic* | for scalars like mass *m* in R^{1}, |

normal | for mathematical symbols like d/dt. |

# position, velocity, acceleration

Sir Isaac Newton was the founder of a gravitational model of our univers. He discovered that matter will apply a gravitational force to all other matter around it.
Thus each particle influences the motion of all other particles. His laws of motion are the three scientific laws which Isaac Newton discovered concerning the
behaviour of moving bodies. These laws are fundamental to classical mechanics.

Newton's first law, the law of inertia or Galileo's Principle, says that
a body's center of mass remains at rest, or moves at a constant velocity in a straight line, unless acted upon by a net outside force.

Newton's second Law, the fundamental law of dynamics, states that
the acceleration of an object of constant mass is proportional to the force acting uopn it.

Newton's third law, the law of reciprocal actions, reveals that
whenever one body exerts force upon a second body, the second body exerts an equal and opposite force upon the first body.

# basic motion

With these three fundamental laws of classical mechanics we can describe the motion of our objects.
These laws provide us with the mathematics to calculate forces between interacting objects.
The laws in mathematical formulae:

1. Law: An object doesn't change its velocity **v** when all forces acting upon it sum up to zero.

(1.1)

**v** = *const.*, or

(1.2)

d**v**/dt = **0**.

2. Law: The acceleration **a** of an object (with constant mass *m*) is proportional to the force **F** acting upon it.

(2.1)

**F** = *m***a**, or

(2.2)

**a** = **F**/*m*.

2. Law in general: without the restriction of constant mass.

(2.3)

**F** = d(*m***v**)/dt.

3. Law: Two interacting objects *A* and *B* exerts an equal and opposite force on each other.

(3)

**F**_{AB} = -**F**_{BA}.

# cross reference to differential maths

There is a simple correlation between position, velocity, and acceleration.

Def.: Given an object with position **x**.

(4.1)

**v** = d**x**/dt.

(4.2)

**a** = d**v**/dt = d^{2}**x**/dt^{2}.

Or, using the defition of momentum and force instead leads to:

(5.1)

*m***v** = *m*d**x**/dt.

(5.2)

**F** = *m***a** = *m*d**v**/dt = *m*d^{2}**x**/dt^{2}.