Guide to Game Development/Theory/Physical motion/Forces

Understanding Sir Isaac Newton

Newton is best known for the apple. The apple hit his head allowing him to have his "eureka" moment about gravity. He was the Lucasian Professor of Mathematics at the University of Cambridge. It is recognised the world around as one of the highest and most prestigious academic posts. Newton is responsible for our understanding of basic motion, forces as well as contributing to optics. He is joint credited for the invention of calculus with Gottfried Leibniz.

Newton's Laws of Motion

Newton's First Law

Newton's First Law - An object in a closed system will remain at constant velocity or at rest unless a force acts upon it.

What the definition stated above means is that if an object was left alone, with no external forces acting on it (that's the closed system part) then it will remain how it is. It will only change velocity when a force has been applied to it.

Newton's Second Law

This is the law that is probably most well known by non-physicists and non-mathematicians.
$F=ma$
However, strictly speaking, the above equation is a branch off of the Second Law.

Netwon's Second Law - The resultant force of an object is directly proportional to the rate of change of momentum of the same object.

What does this mean? Well you need a basic knowledge of calculus and to have read up on suvat's. The rate of change of momentum basically means the differential of momentum with respect to time. Where it says "Resultant Force" it means the overall force of a system. If a car is dragging a trailer, the overall force is calculated using this law. The following derivation follows steps on how to get to the renowned $F=ma$ .

 Momentum is given by  $p$ . 


 $F\propto {\frac {\Delta p}{\Delta t}}$ 


 $F={\frac {\Delta p}{\Delta t}}$ 


 $\Delta p=m(v-u)$ 


 $F=m{\frac {(v-u)}{\Delta t}}$ 


 $a={\frac {v-u}{\Delta t}}$ 


 $F=ma$ 

Provided that the object accelerates uniformly, this equation works.

Newton's Third Law

Newton's Third Law - Every action has an equal and opposite reaction.

This is what the law is in layman's terms. The law is talking about forces in physics rather than actions and reactions in chemistry. This law is what gives rise to "normal forces", or "reaction forces". These forces are directly opposite to the initial force.

If object A exerts a force on object B, then object B will exert an equal and opposite force on object A.

So what does this mean? Imagine that "Object A" is a cup on a table and the table was "Object B". When you look at the cup, it just sits on the table. We know that the cup has a force acting down due to gravity, so why isn't it falling through the table? This answer is that the table is exerting a force that is equal to the weight of the cup but in the opposite direction. This opposite force is the electrostatic repulsion of electron in atoms but we won't get into detail with particle physics. All you need to know is

The cup has a weight acting down
The table has a reaction force to the cup's weight that is acting directly opposite to the cup's weight.

Newton's Laws in Practice

Resolving forces

Equilibrium - When all forces cancel each other out so that they are balanced.

An example of equilibrium would be when there are two forces on the left (10N and 20N) and there is one force on the right (30N). 10N + 20N = 30N.

Resolving a force - This is where you have a force that's being acted on at an angle, and you split the forces into different components (e.g. splitting it into a force in x and splitting it into a force in y).

Example of resolving forces in 2D and 3D

To find the weight/downwards force of an object you need to use:

$Weight=mass*gravity\rightarrow W=mg$

If the object is resting on a surface, then there is an upwards force known as the normal reaction force, this force is equal to the weight.

A surface has a coefficient of friction( $\mu$ ) which will greater than or equal to 0 (In reality a surface won't be exactly 0, but for simulation it's fine to say it is if it's close enough/if friction should be ignored). The resistance force is calculated by:

$Resistance=\mu R$

Top part of the image: W = mg = R Bottom part of the image: Resolving the weight on a slope.