THEORY I: OPERATING FORCE-VELOCITY THEORY OF M4NA FRAMEWORK
Author: Shah Ahmed Fazley Rabbi, BAM, ndc
Framework: Mechanics for New Age (M4NA) Classical Mechanics
Created: 14 June, 2026
THEORY I: OPERATING FORCE-VELOCITY THEORY
ABSTRACT
The Mechanics for New Age (M4NA) framework proposes a quantitative description of mechanical operations centred on the direct correspondence among the value of applied/operating force, velocity, duration, accumulated applied-force quantity, and actual travel. The framework distinguishes explicitly between the value of applied force, denoted by f and its quantity, denoted by F. Within the proposed formulation, the value of applied/operating force corresponds directly with velocity, expressed as $f\propto v$ or: $f=kv$, where k is the universal proportionality constant applicable throughout an ongoing independent applied-force operation.
The framework further defines the quantity of applied force through duration: $F=ft$, while actual travel during an interval is represented by: $s=vt$. Consequently, the corresponding quantity–travel relationship is expressed as: $F_i=ks_i$ and for the complete operation, $F=kS$.
The M4NA framework therefore treats an operation as an interconnected sequence in which applied/operating force produces velocity, velocity corresponds to actual travel and duration converts operating values into accumulated quantities. Changes in the value of applied force are correspondingly reflected in velocity, while the duration associated with each value determines its contribution to the total quantity and total travel.
The framework is developed through six interconnected theories: (I) Operating Force-Velocity Theory, (II) Operating Quantity-Actual Travel Theory, (III) Force-Travel Proportionality Theory, (IV) Pattern Correspondence Theory, (V) Interval-to-Total Consistency Theory, and (VI) Duration and Quantity Theory. Together, these theories provide an interval-by-interval and complete-operation formulation in which: $f_i=kv_i$, $F_i=f_i t_i$, $s_i=v_i t_i$, $F_i=ks_i$, and: $F=kS$.
A central feature of the framework is its distinction between quantity and pattern. Different temporal patterns of force values and velocities may produce the same total quantity or total travel, while different durations may produce different totals even when corresponding values are the same. Thus, equality of accumulated quantities does not imply equality of operational histories.
The framework also maintains that the universal k remains applicable throughout an ongoing operation. When an additional applied force is introduced during that operation, it becomes part of the resultant applied/operating force rather than constituting a separate operation or requiring a new proportionality constant. The resulting change is represented through the corresponding force value and velocity.
The six theories are therefore intended to constitute a single internally connected theoretical structure rather than six independent propositions. Their combined formulation provides a quantitative language for describing mechanical operations at the levels of value, quantity, interval, duration, velocity, actual travel, pattern, and complete operation.
SECTION 1
DEFINITION
The Operating Force-Velocity Theory establishes the primary M4NA relationship between the value of applied/operating force and the velocity produced during an operation.
In M4NA, operating force is applied force. When more than one applied force acts during an operation, their combined result constitutes the resultant applied/operating force. The value of that applied/operating force is denoted by f. The theory proposes that the value of applied/operating force corresponds directly with velocity. Its fundamental mathematical form is: $f\propto v$ or: $f=kv$, where:
f= value of applied/operating force;
v= corresponding velocity;
k= universal proportionality constant applicable throughout the ongoing operation.
SECTION 2
CENTRAL PROPOSITION OF THEORY I
The Operating Force-Velocity Theory may be stated as follows: The value of applied/operating force corresponds directly with the velocity produced during the operation: a greater applied-force value produces a greater corresponding velocity, a lower applied-force value produces a lower corresponding velocity and a constant applied-force value corresponds to a constant velocity. In symbolic form: $f_1>f_2\Rightarrow v_1>v_2$, $f_1<f_2\Rightarrow v_1<v_2$ and: $f=\text{constant} \Rightarrow v=\text{constant}$. The reverse observational correspondence may also be used: $v_1>v_2\Rightarrow f_1>f_2$ and: $v_1<v_2\Rightarrow f_1<f_2$. The important point is that force value and velocity are proposed to correspond directly, rather than being treated as unrelated quantities.
This direct correspondence constitutes the central proposition from which the subsequent quantitative relationships of the M4NA framework are developed.
SECTION 3
MATHEMATICAL FORM
The direct proportionality may be expressed as: $f\propto v$. Introducing the proportionality constant: $f=kv$. Therefore: $k=\frac{f}{v}$ for a nonzero corresponding force value and velocity. For an ongoing independent operation, k remains the same throughout that operation. Thus, for successive intervals: $f_1=kv_1$, $f_2=kv_2$, $f_3=kv_3$ and so forth. Consequently: $$\frac{f_1}{v_1}=\frac{f_2}{v_2}=\frac{f_3}{v_3}=k$$. The relationship therefore applies not merely to one selected point but throughout the represented operation.
SECTION 4
DIRECT CORRESPONDENCE
The expression f∝v means that changes in the value of applied force correspond directly to changes in velocity.
Increasing applied-force value: If: f_1<f_2<f_3, then: $v_1<v_2<v_3$.
Decreasing applied-force value: If: f_1>f_2>f_3, then: $v_1>v_2>v_3$.
Constant applied-force value: If: f_1=f_2=f_3, then: $v_1=v_2=v_3$.
Thus, the value pattern of applied/operating force corresponds directly with the velocity pattern.
SECTION 5
REVERSE CORRESPONDENCE
Because: f=kv, the relationship may also be expressed as: $v=\frac{f}{k}$. Therefore, when k remains constant, velocity may be used to identify the corresponding applied-force value. If: v_1>v_2, then: $f_1>f_2$. If: v_1<v_2, then: $f_1<f_2$. The relationship is therefore bidirectional at the level of correspondence: $f↔v$. This does not mean that force value and velocity are identical quantities. They remain distinct M4NA concepts linked by the proportionality relationship.
SECTION 6
ZERO-VALUE BOUNDARY
The direct proportionality also establishes a zero-value boundary. From: f=kv, when: v=0, then: $f=0$. Conversely, when: f=0, then: $v=0$. Therefore, within the M4NA formulation: $f=0\Longleftrightarrow v=0$. This expresses the M4NA position that a zero value of applied/operating force corresponds to zero velocity at the relevant instant. Likewise: $v\ne 0\Rightarrow f\ne 0$. This zero boundary follows directly from the proposed proportional relationship rather than being introduced as a separate assumption.
SECTION 7
INTERVAL-BY-INTERVAL FORM
For an operation divided into intervals: i=1,2,3,…,n, the theory applies separately to every represented interval: $f_i=kv_i$. Thus: f_1=kv_1, f_2=kv_2, … f_n=kv_n. The same k remains applicable throughout the ongoing operation. This allows an operation to contain: uniform velocity; increasing velocity; decreasing velocity; mixed velocity patterns. The corresponding applied-force value follows the same pattern.
SECTION 8
NUMERICAL ILLUSTRATION
Consider six equal intervals with: v_i=2,4,6,6,4,2. Let: k=2. Then: f_i=kv_i gives: f_i=4,8,12,12,8,4. The two sequences are therefore: $v_i=(2,4,6,6,4,2)$ and: $f_i=(4,8,12,12,8,4)$. The increasing portion of the velocity pattern corresponds to an increasing applied-force-value pattern; the constant portion corresponds to a constant applied-force-value pattern; and the decreasing portion corresponds to a decreasing applied-force-value pattern. This is a mixed-phase operation, but it remains governed by the same k.
SECTION 9
CONSTANT VALUE AND CONSTANT VELOCITY
Suppose: f=8 throughout an operation and: k=2. Then: v=f/k gives: $v=4$. Thus: f=8,8,8,8,8,8 corresponds to: v=4,4,4,4,4,4. A constant applied-force value therefore corresponds to a constant velocity within the M4NA relationship.
SECTION 10
A PATTERN THAT MATTERS
Consider: f_i=8,8,8,8 while: v_i=2,4,6,8. The applied/operating-force value is constant: f_i=”constant”. But velocity changes: v_i↑. As like Newtonian mechanics and descriptions. This illustrates a fundamental difference between the M4NA Operating Force-Velocity Theory and the Newtonian framework.
10.1. M4NA assessment
Under the M4NA Operating Force-Velocity Theory, f_i∝v_i. Therefore, the value of applied force and velocity are directly corresponding quantities. If velocity changes, 2→4→6, the corresponding value of applied force must also change. Consequently, the pattern: f_i=8,8,8 while v_i=2,4,6 cannot occur within the M4NA operating relationship. The corresponding ratios would be: $\frac{f_1}{v_1}=4$, $\frac{f_2}{v_2}=2$, $\frac{f_3}{v_3}=\frac{4}{3}$. Thus, there is no single constant ratio k.
10.2. Newtonian assessment
The same numerical pattern can nevertheless be represented within the Newtonian framework because Newtonian mechanics does not impose the M4NA condition: f_i∝v_i. In the Newtonian formulation, F=ma, force is related to acceleration rather than directly to velocity. Therefore, a constant force can be associated with a changing velocity when the acceleration remains constant. For example, with constant mass and constant force, Newtonian mechanics permits constant acceleration and consequently a changing velocity. Thus, the pattern that is excluded by the M4NA Operating Force-Velocity Theory is not excluded merely because it is numerically impossible. It is excluded because it conflicts with the specific force-velocity correspondence proposed by M4NA, whereas the Newtonian framework does not contain that same direct correspondence requirement.
10.3. The direct test
The distinction can therefore be tested very clearly:
Test condition M4NA Newtonian framework
v_i changes f_i must correspondingly change Constant force may coexist with changing velocity
f_i=8,8,8, v_i=2,4,6 Not permitted by Theory I Can be represented
Direct f-v correspondence Required Not a fundamental requirement
Primary force relation f∝v F=ma
The significance of this test is therefore straightforward: A numerical pattern that can be accommodated by the Newtonian framework but cannot occur under the M4NA Operating Force-Velocity Theory provides a direct experimental question for distinguishing the two frameworks.
The appropriate empirical test is not to ask which framework can write the numbers down. It is to determine whether a real physical operation can produce the pattern.
SECTION 11
ADDITIONAL APPLIED FORCE DURING AN OPERATION
The theory does not treat an additional applied force introduced during an ongoing operation as automatically creating a new operation.
The additional force becomes part of the resultant applied/operating force.
Accordingly, the resulting change is represented by the corresponding value f_i and velocity v_i. The same universal k remains applicable: $f_i=kv_i$.
Thus, if an additional applied force causes the resultant applied-force value to increase, the corresponding velocity increases in accordance with the same relationship. No separate proportionality constant is introduced merely because the resultant force has changed.
SECTION 12
RELATION TO DURATION AND QUANTITY
Theory I concerns the value level of the operation. Duration then converts those values into quantities. For each interval: $F_i=f_i t_i$ and: $s_i=v_i t_i$.
Substituting: f_i=kv_i, gives: F_i=(kv_i ) t_i and therefore: $F_i=k(v_i t_i)$. Since: s_i=v_i t_i, we obtain: $F_i=ks_i$. Thus, Theory I provides the value-level foundation for the quantity–travel relationships developed in the subsequent theories.
SECTION 13
FROM INDIVIDUAL INTERVALS TO THE COMPLETE OPERATION
For the complete operation: $F=\sum_{i=1}^{n}F_i$ and $S=\sum_{i=1}^{n}s_i$. Since: F_i=ks_i for every interval and the same k applies throughout the operation: $F=\sum_{i=1}^{n}ks_i$. Therefore: $F=kS$. Theory I consequently provides the first link in the chain that leads from the value relationship:$f=kv$ to the complete-operation relationship: $F=kS$.
SECTION 14
DIMENSIONAL TREATMENT
Under the finalized M4NA dimensional formulation: [f]=m/s and: [v]=m/s. Therefore: f=kv requires: $[k]=1$. Thus k is dimensionless within the M4NA system. This dimensional treatment is part of the M4NA formulation and is not be replaced by a conventional force-unit interpretation of f.
SECTION 15
THEORETICAL CONSEQUENCES
Theory I establishes several direct consequences.
15.1 Greater force value: f↑⇒v↑.
15.2 Lower force value: f↓⇒v↓.
15.3 Constant force value: f=”constant”⇒v=”constant”.
15.4 Greater velocity: v↑⇒f↑.
15.5 Lower velocity: v↓⇒f↓.
15.6 Zero boundary: f=0⟺v=0.
15.7 Pattern correspondence: The temporal pattern of f_i corresponds directly to the temporal pattern of v_i.
These are consequences of the central proposition: $f\propto v$.
SECTION 16
THEORETICAL SIGNIFICANCE
The significance of Theory I lies in establishing the primary value-level relationship of the M4NA framework. The theory does not begin with accumulated quantity or total travel. It begins with the relationship between the value of the applied/operating force and the velocity corresponding to that operation. The structure is therefore: $\text{Applied/Operating Force Value} \longrightarrow \text{Velocity}$. Once duration is introduced: $f\longrightarrow ft=F$ and: $v\longrightarrow vt=s$. The relationship can then be extended to accumulated quantities and complete travel. Theory I is therefore the foundational value-level theory from which Theories II–VI develop the quantity, travel, pattern, interval-to-total, and duration relationships.
SECTION 17
FINAL FORMULATION
The Operating Force-Velocity Theory may finally be stated as: The value of applied/operating force corresponds directly with the velocity produced during the operation. A greater applied-force value corresponds to a greater velocity, a lower applied-force value corresponds to a lower velocity and a constant applied-force value corresponds to a constant velocity. The reverse correspondence also applies: greater velocity corresponds to greater applied-force value and lower velocity corresponds to lower applied-force value. Its fundamental mathematical expression is: $f\propto v$ or: $f=kv$. For every interval of an ongoing operation: $f_i=kv_i$. The same k remains applicable throughout the operation, including where additional applied force becomes part of the resultant applied/operating force. Through duration: $F_i=f_i t_i$ and: $s_i=v_i t_i$, leading to: $F_i=ks_i$ and ultimately: $F=kS$. Thus, Theory I establishes the first and most fundamental quantitative correspondence in the M4NA framework: $\text{Applied/Operating Force Value} \longleftrightarrow \text{Velocity}$.