Note on update July 2nd: I had to make an update to this post as I had erred in some of the equations. The conclusions are the same but now the formulas are correct. Mainly, I had neglected to balance the force equations on both side which resulted in much to-do in the twitter realm.
Before I get into the rational explanation for the falsification of terminal velocity, I need to make it clear what I’m not saying. My experience with posting controversial ideas tends to make one focused on the negative response rather than the positive.
I’m not saying that their isn’t a maximum velocity that objects in free fall achieve. This is experimentally demonstrable and not the point of this post. What I am saying is that the explanation for this phenomenon is incorrect and the mathematical description is invalid.
The constant of proportionality, G, is the gravitational constant. Colloquially, the gravitational constant is also called “Big G”, for disambiguation with “small g” (g), which is the local gravitational field of Earth (equivalent to the free-fall acceleration). The two quantities are related by g = GME/r2 (where ME is the mass of the Earth and rE is the radius of the Earth).
The supposed local value for gravity on the earth is g = GME/r2 which translates to 9.8 m/s². For the purposes of this proposal, small ‘g’ is sufficient.
The current explanation for terminal velocity as per wiki:
Terminal velocity is the highest velocity attainable by an object as it falls through a fluid (air is the most common example). It occurs when the sum of the drag force (Fd) and the buoyancy is equal to the downward force of gravity (FG) acting on the object. Since the net force on the object is zero, the object has zero acceleration.
There is a significant issue with this explanation which ties into the invalid structure of the mathematical derivation.
According to the standard model, the only downward force that is acting on the object is the force of gravity and any opposite force will result in the reduction of the downward force of gravity itself. Objects can only accelerate in free fall; they can never achieve a constant velocity. Again, this is a rebuttal to the current explanation of terminal velocity not a dismissal of terminal velocity itself.
If the net force acting on the object is zero, then the object will simply stop accelerating and since the only downward force is an acceleration, the object should stop moving (akin to neutral buoyancy). Any objection to this conclusion requires either the addition or subtraction of forces. For example, some will argue that once the net forces are in balance the final velocity achieved at that point, is terminal velocity. They will also pull out Newton’s 1st law:
An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force
We need to take apart all the assumptions about a free falling object in order to see why terminal velocity as it is currently understood doesn’t work. As mentioned before, a free falling object only has the force of gravity acting on it therefore the speed is an acceleration not a constant velocity so once the acceleration is zero due to drag, it should simply slow down to zero. If there was no unbalanced force of drag, the object would continue to fall at an increasing rate. A simple analogy: two cars tied together and accelerating in opposite directions would go nowhere but the forces are still active. There are no additional forces present. The cars wouldn’t just start drifting at a constant velocity towards one car or the other.
So where does the velocity value even come from? To figure that out we need to see how it is derived (from Wiki):
Derivation for terminal velocity
Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation):
At equilibrium, the net force is zero (F = 0);
Solving for v yields
If you examine this formula closely you will see where the error creeps in. It’s not that the math itself is incorrect, but the conclusions are incorrect relative to the math.
There are two distinctive forces present in this formula – gravity and drag. The portion of the equation that describes the force of gravity is “mg”. The second portion of the formula is “1/2 pv²AC” which describes the drag forces. This is the upward force due to drag which counter-balances the supposed force of gravity.
To solve for “v” we need to isolate it like this:
mg = 1/2 pv²AC (multiply both side by 2)
2mg = pv²AC (divide both sides by pAC)
2mg / pAC = v² (get the square root of each side)
v = √ 2mg / pAC
The variable “v” is a function of the upward force of drag so it cannot be isolated out in real life. It’s part of the accelerating upward force. From a mathematical point of view, it simply describes the upward velocity of the air at a specific point in time. But for the equation to balance out we need to also take the square root of the force of gravity and the mass of the object. That doesn’t even make sense. What does the square root of mass even describe in real life?
So even though the math works (meaning it balances out) it has nothing to do with the actual events we would observe. The fact that the resultant force is called a velocity rather than an acceleration is also interesting since the downward force on any object at anytime is gravity (which is an accelerating force) according to the standard model.
Secondly, the article goes on to state that buoyancy effects can be subtracted from the mass of object.
Buoyancy effects, due to the upward force on the object by the surrounding fluid, can be taken into account using Archimedes’ principle: the mass has to be reduced by the displaced fluid mass , with the volume of the object. So instead of use the reduced mass in this and subsequent formulas.
This is in direct conflict with the concept of mass since the mass of an object is always constant; it is the supposed force of gravity that gives the object weight so it must be the weight that is displaced NOT the mass of the object. From Wiki:
Assuming Archimedes’ principle to be reformulated as follows,
then inserted into the quotient of weights, which has been expanded by the mutual volume
yields the formula below. The density of the immersed object relative to the density of the fluid can easily be calculated without measuring any volumes:
(This formula is used for example in describing the measuring principle of a dasymeter and of hydrostatic weighing.)
According to the standard model, the weight of an object is equal to the mass times the force of gravity:
There are two important factors that need to be considered when looking at drag and buoyancy on a falling object.
In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid. The formula is accurate only under certain conditions: the objects must have a blunt form factor and the system must have a large enough Reynolds number to produce turbulence behind the object.
So the key to drag is turbulence. This is one of the effects that slows the object down as it falls. Turbulence also increases over time. There is zero turbulence when object is suspended in air and motionless and an increasing turbulence over time once the object is released.
Of particular importance is the dependence on flow velocity, meaning that fluid drag increases with the square of flow velocity. When flow velocity is doubled, for example, not only does the fluid strike with twice the flow velocity, but twice the mass of fluid strikes per second. Therefore the change of momentum per second is multiplied by four. Force is equivalent to the change of momentum divided by time.
As the object falls, the flow velocity (meaning the total amount of air pushing back on the object) and subsequently the total mass of air will increase. The result is an increasing increase in turbulence behind the object. The total turbulence is directly proportional to the maximum velocity an object can sustain in free fall minus any buoyancy effects.
So the proper formula for buoyancy effects would be mg – pV or W – pV. Therefore, the proper formula for terminal velocity would be the weight of an object minus buoyancy effects minus drag or v = [W – (pV)] – (1/2 u²pAC). We can see that this is simply a momentum calculation that subtracts two opposing effects – buoyancy and drag.
The reason for an increasing increase of velocity (acceleration) during the initial release of the object is due to the turbulence requiring time to increase. But as the turbulence increasingly increases to a maximum value, the object begins to slow down from it’s initial acceleration in direct proportion to the value of turbulence.
We could postulate that the acceleration of an object in free fall would be unlimited if turbulence was zero. Therefore, the acceleration of an object due to the force of gravity does NOT have to be evoked to explain an object in free fall but only the difference between the density of air and the falling object. It is the turbulence that is slowing down the object. Once the density of the falling object meets the density of the ground, the falling stops. Thus, there is no downward force acting on the object just a difference in density.
If there was indeed an accelerating force due to gravity, the object on the ground should continue to move towards the center of the earth. Why should the acceleration due to gravity stop just because air gave way to ground? The fact that we do not continue to accelerate towards the center of a ball when, by all theoretical and mathematical requirements, we should be, is a clear falsification of gravity.
Gravity – an imaginary force
I further postulate that the mass and the weight of an object are the same thing and gravity is an imaginary force that is only applied to imaginary celestial objects (planets and other imaginary celestial objects must all be in free fall around some other object). By direct observation, an object at rest on the surface of the earth is no longer subject to the imaginary force of gravity since the weight of an object must continually increase due to the constant acceleration.
If Newton’s third law is to be followed, the object “at rest” (but not really) on the surface of the Earth must be continually pushing back with an “equal and opposite reaction” over time since an acceleration is the change of velocity over time. So with each second the net force being applied to the object must increase. Let’s take an example from Khan Academy:
What does the area represent on an acceleration graph?
The area under an acceleration graph represents the change in velocity. In other words, the area under the acceleration graph for a certain time interval is equal to the change in velocity during that time interval.
It might be easiest to see why this is the case by considering the example graph below which shows a constant acceleration of 4 m/s² for 9s.
If we multiply both sides of the definition of acceleration, a = Δv / Δt , by the change in time, Δt, we get Δv = aΔt.
Plugging in the acceleration 4 m/s² and the time interval 9s we can find the change in velocity:
Δv = aΔt = (4m/s²)(9s) = 36 m/s
Multiplying the acceleration by the time interval is equivalent to finding the area under the curve. The area under the curve is a rectangle, as seen in the diagram below.
The area can be found by multiplying height times width. The height of this rectangle is 4 m/s², and the width is 9s. So, to find the area also gives you the change in velocity.
area = (4m/s²)(9s) = 36 m/s
The area under any acceleration graph for a certain time interval gives the change in velocity for that time interval.
We can now take the acceleration due to gravity (9.8 m/s²) and apply the same model. A person standing on the surface of the Earth must experience a constant change in velocity:
Δv = aΔt = (9.8m/s²)(1s) = 9.8 m/s
Δv = aΔt = (9.8m/s²)(2s) = 19.6 m/s
Δv = aΔt = (9.8m/s²)(3s) = 29.4 m/s
Δv = aΔt = (9.8m/s²)(∞s) = ∞ m/s
According to the standard model of gravity, a person in free fall is weightless, therefore their mass is not subject to measurement. Mass as a function of weight is only applicable when there is resistance to counteract free fall. As the mass of the person begins to impact the surface of the Earth, they begin to weigh something and that mass is subject to an increasing velocity which translates into an increasing weight.
How do we measure weight? This can be measured on a standard scale. A scale has springs that can compress and translate the downward pressure into a weight. As long as the springs can handle the increasing weight, so will the person increase velocity and we can measure the increase. But the force of the mass of a person is dependent upon the velocity at a particular time since:
tΔp = tmΔv
Of course this is not a common way to express momentum since it is a change in momentum due to a change in velocity. However, in this case, the change in momentum is being translated into a change in weight:
If “Δvm = maΔt”
Then “ΔtW = mΔv” or “ΔtW = maΔt”
So what is a kilogram or a pound? It is an arbitrary unit of measurement based upon various assumptions. The primary assumption is that weight is a function of gravity and mass. However, we can clearly see that if velocity is constantly changing due to the force of gravity then the resultant weight must increase. It is required by the standard model of gravity that the mass and acceleration of the object are fixed and do not change. It’s the velocity that must be changing over time.
The objection to this proposal is that time cannot be a function of Weight or of Force. The standard way to express the equation is F = ma. However, this equation contains within it a time function as ma = Δv / Δt · m. This results in:
Δvm = ΔtF
ΔtF = maΔt = Δvm
Therefore, if there is no change in velocity, as anyone can observe, then we are left with:
F = W = m
So what is mass? How does something weigh more than something else? That is our next topic.
The Foundation of Mass
What we are left with are two attributes of mass – Density and Buoyancy. Gravity, in this instance, is no longer an accelerating force inherent to the mass of an object but a synonym for mass itself. There is no intrinsic force attributable to mass.
Let’s start with a quote from Newton himself:
“It is inconceivable that inanimate Matter should, without the mediation of something else, which is not material, operate upon, and affect other matter without mutual contact. Gravity should be innate, inherent and essential to matter, so that one body may act upon another at a distance thru a vacuum, without the mediation of any thing else, by and through which their action and course may be conveyed from one to another, is to me so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into (for) it. Gravity [mass] must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I have left to the consideration of my readers.” – Sir Isaac Newton, Letters to Bentley, 1692
We are indeed considering his words. What we can extract from this is that gravity or mass is the result of something not the cause of something and a “mediating” entity must be involved.
Here is a quote from Ken Wheeler:
Before embarking on explaining magnetism, first the all important word field must be rigorously and scientifically defined with ambiguity and misconceptions fully removed. A field (Greek: χώρα) is a conjugate nonspatial attribute between the subject, Ether, and the object causing the perturbation (or EM induction), either as gravity [weight/mass], electricity, magnetism, or electromagnetism. A field has relevance only as a relational perturbation between the subject and the object causing its appearance; a field is specified only as regards to: electricity, magnetism, gravity, matter, and dielectricity (and another unmentioned for another article); however as it must be necessitated electricity, dielectricity, gravity [weight/mass], and magnetism are by their very principle not different from the Ether itself. The very term ‘magnet’ merely denotes the electrified mass formerly ‘not-a-magnet’; as such magnetism and magnet are abstractions or distinctions without a difference except as relates to the coherent field charges of the before and after mass. Conceptual disambiguation between a magnet and magnetism itself cannot be enjoined, and is merely a fallacious reification of connotative abstractions.
Ken Wheeler: Uncovering the Missing Secrets of Magnetism p.27
Everything seems to come down to the historically dismissed Ether. It must be noted that even the high priest of gravity, Newton, proposed an Ether. The persistent objection to the dismissal of gravity is always, “what is the alternative”? That an empirically based science that includes the Ether is a valid alternative and has always been a valid alternative seems to elude those so certain of gravity. It’s the implications of an Ether that frightens them so.
I will not put words into Mr. Wheeler’s mouth so it is important to note that he does not dismiss Gravity per say but the origins of it. Based upon Mr. Wheelers writings, I can safely say that he is NOT a proponent of the Flat Earth so I can only assume that he believes in planets, comets, gravity, etc. So in no way am I linking him to the Flat Earth theory. But I digress…
Current scientific thought requires gravity to be an inherent property of mass itself rather than an affect of a mass within the Ether (like the waves on an ocean is the ocean but not inherent to the ocean; something else causes waves). If there is NOT an intrinsic force to mass then much of modern astronomy falls apart.
Density and Matter
So what causes mass? In mathematical terms it would be p * V = m (density * Volume = mass). In this sense, mass is a function of density and volume. We know that volume is a function of geometry so what is density?
Magnetism is purely radiative, is the termination of electrification and the end-of-road byproduct of dielectricity. Dielectricity comes before everything else in the four-part schema of Force Unification. Dielectricity and magnetism are the two co-principles of the universe. So how do you get magnetism out of dielectricity, since magnetism requires a subject to emanate from or itself is the termination point of either mass in movement or electricity as it terminates? The answer is that dielectricity terminates into the creation of matter, which itself then has in this conjugate relationship, magnetism as its radiative principle (the proton as found in hydrogen, the most abundant element is magnetically dominant, is the polarized charging dynamo for its discharge plane of interatomic magneto-dielectric volume). Creation (dielectricity) and radiation (magnetism), and their two byproducts, electricity and mass, or gravity [density] as centripetal attributes choate to mass / matter.
Ken Wheeler: Uncovering the Missing Secrets of Magnetism p.38
What Ken Wheeler is calling “gravity” in this instance would simply be the density attribute of mass. A dielectric object (not-a-magnet mass) has a specific density due to the “centripetal attributes choate” to it. It is the centripetal/centrifugal attributes of the object that ultimately give objects their density and subsequently makes objects heavier or lighter than others.
A complete thesis developed and proven by Ken Wheeler is available for anyone to read. It is incumbent upon the read to disprove his empirically based science before dismissing the concept of an Ether. This is true science and if you fancy yourself a true scientist or a rational human being, you will need to apply the proper scientific rigor rather than falling into Scientism consensus.