Said already earlier, that Newton missed vertical systems and moves. He refined Galilei's gravity researches to laws.
The old Archimedes law about liquids, goes to gases with small changes.

I was making ball simulation for ATS Roulette game. Tried to calculate kick off energies for ball. Realized that ball's acceleration - during the moment, the push ends - is zero. 

During the moment ball takes off F_k = ma results to plain zero. This cannot be true. The ball is certain to have kinetic energy, with what it rolls. It cannot roll without kinetic energy.

When I puzzled with the issue, came to the conclusion that kinetic energy of an object is actually speed-weigh system at moment X.

With acceleration you calculate speeds at moment X, but actual energy must come from current speed. When freely rotating ball slows down, ma calculation always ends into zero. When you use acceleration for calculating new speed, F_k = mv² results into new kinetic energy for the object.

F_k = Objects kinetic energy at moment X
m = mass * G
v = speed at moment X

Power of two is needed for keeping the old Newton with the formula. This formula cannot be correct. Native unit for Newton is kg m/s². Power of 2 results into m²/s² 

Ooops. ... In the actual calculations for balls I used air's density for correcting the unit. Density is included in full formula for air resistance force. Unit for density is kg/m³. When you for example calculate air resistances for the balls, the formula with Av energy results into incorrect unit. 

Fk = A * v2 * q Fk = kinetic energy at moment X, N - kgm/s2 q  = density of the surrounding gas/liquid, kg/m3 A  = frontal area, m2 v  = speed at moment X, m/s m2 * m2 / s2 and kg / m3 => kg m/s2

- In this formula you do not use mass. Mass affects to the acceleration. You use the mass for calculating changes in speed.
- Dense material results in to bigger kinetic energy. When your roll your balls in thick gas, you need more force for keeping the speed.
- When you have falling object, you calculate gravitational mass-speed system, and add it to your force calculation.
- New formula works also in space, where you do not have gravity. You can use it when you calculate predicted acceleration, braking and steering forces.
- on earth speed is known to operate with curved force system.

Gas Extension to Archimedes

When object moves in gas, it follows Archimedes law about liquids. The object takes it's volume from the gas. The difference is that, gas shrinks and liquid escapes, when object comes.

The shrinking gas creates a pressure system around the intruder. When you go outdoors, and stop. The pressure system spreads into surroundings. When you move, you create new pressure systems around yourself all the time. When you are in closed space, the pressure system cannot spread, it remains.

It does not matter much, whether object moves or it is blown. It is easier to keep your momentary place, when you stand in the typhoon winds. Walk in hurricane winds is more difficult : Standing object has usually some stable parts in the surrounding pressure system.

In gas captured volume creates pressure systems to all available directions.

It is well known, that air resistance factor comes from turbulences around object. The resistance remains relatively stable, until some major part in the front starts to create bigger turbulences. Turbulences creates buffers to front of the moving object. Moving object itself creates only pressure systems to the front.

With aerodynamic front, you can slow down the creation of pressure. With tilted top and bottom you redirect part from the pressure to the top or bottom of the object. You cannot redirect pressure away from the object. Moving object must always handle the pressure, which comes from the frontal side projection.

By default gas atoms do not change their relative positions, when object comes. The grid stretches and shrinks as whole. The atoms start to move, and the grid gets broke, when you have too much speed and area. Quite obvious that badly broken grid creates the turbulences to the air / gas.

In order to get air in and out from combustion engine's cylinder, you must break the atom grid with speed. You cannot do such with pressure only. Propel for example moves atoms with it's wings.

It is known that you can move atoms and objects with flow, without turbulence. Turbulence could come from the broken movement system. When so turbulence is born, when long threads, with what atoms moves, gets broke : Atoms do not keep the marching order, they search new locations from where-ever they can.

In planes the captured volume creates red pressure system  to above. Green to below.
Green pressure system creates the force, which keeps the plane in the air.
The big red arrow denotes planes weight, and gravitational force. Car shares the system.

From the above can say, that objects ... like cars and planes ... would need measured cw curves for various speeds. Speed increases the amount of turbulunces. There obviously are some key points, when for example windshield starts to create turbulences.

When you test cars and planes in wind tunnel, you should find a speed, when shape loses it's aerodynamics. Air becomes a turbulent mess. You could inform the constructional speeds for the shapes and bodies. ( Idea / proof comes from pressurized gas and oil tubes. Electric wire has similar kind of electric power limit. )

You can never measure cw factor with 3D model. Air resistance factor must be tested in wind tunnel or nature. Pressure system around moving object is too complicated for computer program. A small change in mirror's mounting leads to total change in the remaining pressure system and so on. In cars, side mirrors are well known to be one of the biggest disturbers of the flow. Mirrors downgrades the overall ( announced ) cw coefficient factor.

Boat with big top speed needs also aerodynamics. All boats need hydrodynamics for the bottom. Hydrodynamics is compilation from aerodynamics and on-ground friction system.

In the picture above you see diagram from bird's over-under pressure system. Flying bird keeps top pressure steady, head and body are straight lined. Bird can create a small vacuum to the top with it's head. After bird has made a hole into air, with it's peak, it uses neck and belly for boosting the lower pressure and increasing the lift. Lower body tilt keeps the pressure under body. Feathers' small threads obviously prevents air from escaping from under the body and wings. Lower-body tilt and feathers allows birds to fly slowly.

Although birds flying system is widely researched, no has measured how "poor" bird's air resistance factor is. You could have done it with killed and filled flying bird.
Long necked bird, like swan,  goes to modified needle shape when it flies. All birds uses belly-body for maintaining the pressure under body. 

Birds are the finest flying things on earth.

Things you find from our nature are hardly ever without some meaning. It is always a big step from mankind and manmade things. Certainty that you are almost 100% sure to find a reason for everything eases science making quite a lot. You do not get lost as easily as you do with manmade things.

Better precision to gravitational friction force ( not yet finished )

Gravitational friction forces are calculated with old formulas. But with bigger precision

When I developed formulas for ATS roulette and car simulation, the current friction force system couldn't provide proper results for the 2D game.

Friction for car comes from the tires. The first thing is to divide cars mass to wheels. Calculation is based on weight center. You divide the mass to wheels with momentum from wheel to weight center. Just like you do in strength calculations. Cars longitudinal and lateral tilt must be included to calculation. Direction from wheel to weight center allows the division to boxes.

When object moves on surface, there are four primary types of forces, which resists the movement. First two forces comes from the raw friction. The wheel and road has unique friction factors. Friction factor comes from smoothness / roughness of surface : Polished, varnished wood has totally different friction than varnished, untreated wooden plank. Material is insignificant. Polished, varnished wood is as shiny and slippery as polished rock.

The second friction force set comes from the softness of object and surface. When object is soft, the area with what the object touches the surface increases. When the surface is too soft, the object falls into surface. The effect of the fall is dependent on objects shape. Fall of rectangular object resists the move. Fall of round object increases the friction force. Fall of rolling round object increases the touchdown area, then it resists the move with the depth of created pit.

The theoretical base for the system is, that ice is always slippery. It does not matter what kind of object there on the ice. So it is possible to define nominal friction factors for various materials. Calculate separate friction forces for both object and surface. Sum them together for getting the total force. Force, which is currently calculated with paired friction factor tables.

When you use nominal friction factors for roughness/ softness, you can build general friction systems. You do not build ready made tables for object surface pairs. You create tables, where surfaces and objects have roughness- softness factors for calculating friction forces. With tables you can calculate areas for touch.

With additional factors you can get more detailed / accurate calculations.

Softness

Softness is conditional factor, it needs a pressure-force system for the beginning and breaking. If object or surface are strong enough to carry the load from object, there aren't any magical, miniature deformations. Nothing ever happens. Some materials gets polished by wear, some gets rougher by wear. Immediate effect is typical for polishing, etc. tools. The effect of these goes to upper atom layers, they are insignificant for rough force calculations.

When pressure from the object is smaller than F1, softness force is zero. When pressure from object is bigger than F2 the underlying object cracks. There are objects, which never bends, there are objects-surfaces, which never cracks. F1 and F2 requires indicator for the mentioned impossibilities.

 

	Case 1 : Object reacts, surface stable In this case the force from object is big enough to shrink / flatten the object, but it is not big enough for surface.  Flat doesn't change length of circle ( much ).  There fore there are two types of flattening. - In small, standard flattening object stretches inside ellipse or stretched rectangle. In this case touch area stays inside ellipse. - Bigger flattening spreads touch-area outside the ellipse / rectangle. Object spreads into surroundings. - Switch in between small and big flattening is smooth. With one look to cars tire, you can verify that flattening takes place in all directions. So you need X-Y factors or graph for softness.
	Case 2 : Object stable, surface reacts In this case the force from object is big enough to dig a pit to surface, but it is not big enough for object. When moving object falls into pit, it has climb up from it. Use red force triangle for getting which is used climbing.  - When angle is 90 degrees angle you use undivided force. - When angle is over 90 degrees angle, you use reversed triangle. The touch area is the bottom of rectangle or area of touching circle / ball.
	Case 3 : Object reacts, surface reacts In this case the force from object is big enough to flatten the object and digging a pit to surface. In simple system you use Cases  1 and 2 for calculating the forces. Precise calculation needs that, you take surface's horizontal push to notice. The pit for object resists the flattening of the object.

Static friction

In principle object-surface does not have static and dynamic frictions. When you think it, it looks like that  static friction is air resistance force. When so it is dependent on objects size and surrounding gas / liquid. You can check how it is, with tests.

Wet / lubrication factors

In lubricated object-surface system the friction grows to the full effect. Friction begins with dry surface factor it ends to lubricants factor. 

On paved road it is easy to notice how the friction vanishes with thickness of ice layer. In the beginning ice keeps the roads roughness, when layer grows thicker the roughness vanishes, it is more and more difficult to keep the car on the road. 

In here partial ice cover on paved road is called black ice. It is invisible to eye, but it is not usually as slippery as visible ice. After certain weather system black ice covers all roads.

Water, snow and ice has many forms. 
- ice, water on ice, snow on ice, etc 
- soft ice, water on soft ice, snow on soft ice, soft ice over ice, etc
- snow, water-ice on snow, ice on snow, frozen snow
- water, puddles, streams and lakes.

All possible forms exits in the nature. During spring time, when sun melts the snow and cold night freezes the melted snow, all forms are possible. During spring, all variations exist in wider scale.

Frozen snow is born, when temperature raises over zero. The topmost snow layer melts. When temperature falls below 0, the snow freezes. You can even drive car over the strongest frozen snow layers. Snow carries load like ice, but it looks like snow. Ice on snow creates thin, hard, mirroring cover over snow. Frozen snow is thicker and it misses the mirror effect. When go and play over frozen ice, there are usually undetectable places with common snow layer. You cannot see the snow pits.

Lateral factor (curve)

Tire belongs to objects, which have different xy friction factors. It needs longitudinal and lateral frictions. In order to draw arc-ellipse for the factor, you need elliptical line, with predefined cross point. With elliptical line and angle you can get correct friction factors and forces.

Shift factors (curve)

Tire which has air space needs another elliptical line for defining the longitudinal and lateral movements movements with various pressures.

Rubble factors and forces

Sand road is an example from the surface, with has loose particles, which starts to roll under wheel when you drive too fast. 

Bounce factors and forces

Fully defined softness needs also bouncing factor. With bounce factor you calculate collision speeds and recovery times for bounce. When you throw soft ball towards hard wall, the ball shrinks. Then it bounces back with a force, it gets from the bounce.