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M4NA Interactive Mechanics Laboratory

Calculate physical quantities, examine interval-by-interval travel, compare numerical patterns, and explore the quantitative relationships presented in the M4NA framework.

How to use the laboratory
Enter the quantities requested in each module and select Calculate. Use the graphs to inspect the numerical results. The calculations are based on the equations displayed in each section. A numerical match does not by itself establish an experimental law.

Part I — Distance and Velocity

Calculate individual quantities and examine accumulated travel.

1Distance Calculator

Enter any two quantities. Leave the quantity to be calculated blank. Values must be non-negative.

s = v × t · F = kS · F/t = k(s/t) · F/t = kv

Distance s (m)

Velocity v (m/s)

Time t (s)

Force quantity F (m), optional

Distance–time graph. The displayed line assumes constant velocity during the calculated interval.

2Average Velocity

Calculate average velocity from total distance and total elapsed time, or calculate a missing quantity.

V = S / T · F = kS · F/T = k(S/T) · F/T = kV

Total distance S (m)

Total time T (s)

Average velocity V (m/s)

Force quantity F (m), optional

Here, V means total accumulated distance divided by total elapsed time. For interval data, use the Interval Travel module to calculate S and T before calculating V.

Part II — Interval Travel and Patterns

Enter a sequence of velocities and durations, then inspect its total and pattern.

3Interval Travel & Pattern Analysis

Each row represents one time interval. The interval distance is calculated from that row's velocity and duration. Total distance is the sum of the interval distances.

Force–distance identity

Force/time–distance/time identity

Force–velocity identity

Fᵢ = ksᵢ

Fᵢ/tᵢ = k(sᵢ/tᵢ)

fᵢ = kvᵢ

F = kS

F/T = k(S/T)

F/T = kV

Interval	Force value fᵢ	Velocity vᵢ (m/s)	Duration tᵢ (s)	Distance sᵢ (m)	Force quantity Fᵢ	Action

Total distance S —

Total time T —

Average velocity V —

Force quantity F —

Velocity–time step graph. Each entered velocity is held constant over its own interval; vertical steps show changes between intervals.

Example: velocities 2, 4, 6, 6, 4, 2 m/s, each lasting 1 second, produce a total distance of 24 m and an average velocity of 4 m/s. The graph shows the entered interval pattern, not a claim about the physical cause of that pattern.

Part III — Applied Force and Travel

Explore the quantitative relationships specified in M4NA.

4Applied-Force Quantity

Enter any two of force value, operating duration, and force quantity. Leave the third field blank.

F = f × t

Force value f (m/s)

Duration t (s)

Force quantity F (m)

The labels follow the M4NA notation supplied for this calculator: f is force value and F is force quantity. These are not conventional SI force units; the page preserves the M4NA dimensional assignments for this calculation.

5Force–Travel Proportionality

Calculate one missing quantity in the relationship F = kS.

F = kS · k = F / S

Force quantity F (m)

Total distance S (m)

Constant k

Graph of F against S for the selected value of k. The line represents the specified mathematical relationship.

In the stated M4NA notation, F and S are both assigned units of metres, so k is dimensionless. A straight-line graph demonstrates the chosen equation; it does not independently verify the physical law.

Part IV — Force–Velocity Pattern

Enter multiple pairs to compare the numerical patterns of f and v.

6Force–Velocity Pattern Analysis

Lock one proportionality constant k for the whole operation. Choose either force values fᵢ or velocity values vᵢ as the sole pattern input; the other pattern is calculated from the same k.

Force–distance identity

Force/time–distance/time identity

Force–velocity identity

Fᵢ = ksᵢ

Fᵢ/tᵢ = k(sᵢ/tᵢ)

fᵢ = kvᵢ

F = kS

F/T = k(S/T)

F/T = kV

Pattern input

Locked proportionality constant k

Interval durations tᵢ (comma-separated)

Input pattern (comma-separated)

The selected input type is the only entered force/velocity pattern. k remains fixed across all intervals and the total operation. The calculator checks the same three identity relationships for every interval and the total: fᵢ = kvᵢ, Fᵢ/tᵢ = k(sᵢ/tᵢ), F = kS, F/T = k(S/T), and F/T = kV.

M4NA Interactive Mechanics Laboratory

Static Observer vs Moving Observer

Examine one train-and-ball operation from two observation positions. The moving observer measures the ball's travel relative to the train. The static observer measures the train's travel and the ball's travel relative to the ground. Both calculations use the same specified time interval.

1. Set the operating conditions

Train velocity (m/s)

Ball velocity relative to train (m/s)

Ball's direction relative to train

Operating time (seconds)

Train starting position on ground (m)

Ball starting position inside train (m)

2. Moving Observer

The observer travels with the train and measures the ball relative to the train.

Ball's travelling distance relative to train

20 m

Ball's relative velocity:5 m/s forward

Operating time:4 s

Ball's final position inside train:20 m

Relative travel: s = |v relative| × t
s = 5 × 4 = 20 m

3. Static Observer

The observer remains on the ground and measures the train and ball relative to the ground.

Ball's travelling distance relative to ground

100 m

Train's travelling distance:80 m

Ball's ground velocity:25 m/s forward

Ball's final ground position:100 m

Ground velocity = train velocity + signed relative velocity
20 + 5 = 25 m/s
Ground travel = |ground velocity| × time
25 × 4 = 100 m

4. Direct Comparison of the Two Observations

This comparison makes the reference position explicit. The ball's relative travel and ground travel are different measurements when the train itself is moving.

Quantity	Moving observer (train reference)	Static observer (ground reference)
Reference position	Train	Ground
Ball velocity used	5 m/s forward	25 m/s forward
Ball travelling distance	20 m	100 m
Train travelling distance	Not measured as movement relative to itself	80 m
Operating time	4 s	4 s

The two observers use different reference positions, so their measured ball-travel distances need not be equal.

5. Compare the Travelling Distances

The bars compare train travel, ball travel relative to the train, and ball travel relative to the ground. These quantities use different reference positions.

6. Compare Positions Over Time

The graph shows the train's ground position, the ball's ground position, and the ball's position relative to the train throughout the selected interval.

7. M4NA Interpretation

The moving observer and static observer measure relative travel from different reference positions.

Moving observer: ball travel relative to train = magnitude of relative velocity × time.

Static observer: train travel = train velocity × time; ball ground velocity is obtained by combining train velocity with the signed ball velocity relative to the train. Ball ground travel = magnitude of ground velocity × time.

Interpretation note: this calculator compares the numerical measurements made from two reference positions over the same time interval. It does not treat the two distance values as if they were the same reference-relative quantity.

M4NA Interactive Mechanics Laboratory

Panel 10 — Compare Newtonian and M4NA Mechanics

Set one six-interval pattern and examine both frameworks side by side. Choose whether velocity values or applied-force values are the pattern input. The M4NA calculation uses one fixed proportionality constant k throughout the operation.

1. Set the comparison conditions

Pattern input

Velocity pattern preset

Newtonian mass (kg)

M4NA proportionality constant k

Newtonian initial velocity (m/s)

Duration of each interval (s)

Six interval velocity values (m/s)

Enter exactly six comma-separated velocity values. M4NA force values are calculated using fᵢ = kvᵢ.

Enter conditions and select Calculate Comparison.

2. Results at a glance

Newtonian calculation

Using mass and interval-to-interval velocity changes

—

Run the comparison to see results.

M4NA calculation

Using fᵢ = kvᵢ and sᵢ = vᵢtᵢ

—

Run the comparison to see results.

3. Six-interval comparison table

Interval	Time tᵢ (s)	Input vᵢ (m/s)	M4NA fᵢ	M4NA Fᵢ=fᵢtᵢ	M4NA sᵢ=vᵢtᵢ	Newtonian force (N)	Newtonian distance (m)
Results will appear here.

4. M4NA identity tests

Check the interval relationships and the total operation using the same fixed k.

Results will appear here.

5. Velocity pattern and accumulated distance

Compare the entered velocity pattern and the cumulative distance generated by each framework.

6. Interpretation

After calculation, this panel summarizes the numerical pattern and identity checks.

The table reports the calculations according to the selected mode and the displayed relationships; the numerical results are not hidden behind a single overall score.

M4NA Interactive Mechanics Laboratory

Static Observer vs Moving Observer

1. Set the operating conditions

Train velocity (m/s)

Ball velocity relative to train (m/s)

Ball's direction relative to train

Operating time (seconds)

Starting position on ground (m)

Ball's starting position inside train (m)

2. Moving Observer

The observer travels with the train and measures the ball relative to the train.

Ball's relative travelling distance

20 m

Ball's relative velocity:5 m/s forward

Operating time:4 s

Ball's final position inside train:20 m

Relative travel: s = v × t
s = 5 × 4 = 20 m

3. Static Observer

The observer remains on the ground and measures the train and ball relative to the ground.

Ball's travelling distance relative to ground

100 m

Train's travelling distance:80 m

Ball's ground velocity:25 m/s forward

Ball's final ground position:100 m

Ground velocity: train velocity + signed relative velocity
20 + 5 = 25 m/s
Ground travel: |ground velocity| × time
25 × 4 = 100 m

4. Compare the travelling distances

The bars compare the distances measured in the two observation positions. They represent different reference positions, not two measurements of the same relative distance.

5. Compare positions over time

The graph shows the train's ground position and the ball's ground position during the selected operating interval. The ball's position inside the train is also shown as a separate reference-relative quantity.

6. M4NA interpretation

The moving observer and the static observer measure different relative travelling distances because they use different reference positions.

Moving observer:ball travel relative to train = relative velocity × time.

Static observer:train travel = train velocity × time; ball ground velocity is obtained by combining the train velocity with the ball's signed relative velocity. The ball's ground travel is the magnitude of its ground velocity multiplied by time.

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M4NA Laboratory Contains Two More Feature. To Explore Please Click Below

Compare Two Mechanics Frameworks

Open the comparison tool in a separate page

Panel 10 — Newtonian vs M4NA

Interactive comparison of force, velocity, travelling distance, interval values and accumulated quantities.

1. Select a pattern and calculation mode

Calculation mode

Velocity pattern

Newtonian mass (kg)

M4NA constant k

Initial velocity (m/s)

Duration of each interval (s)

Enter target velocities at the end of each interval (m/s)

Interval 1

Interval 2

Interval 3

Interval 4

Interval 5

Interval 6

Velocity-first mode: the entered values are the target velocities at the end of the six intervals. Newtonian force is calculated from the velocity change; M4NA force value is calculated from f = kv.

2. Results at a glance

Newtonian Mechanics

Force value
F = ma

Final velocity —

Force values (intervals) —

Final interval force value —

Accumulated distance —

Average velocity —

Newtonian force impulse —

Newtonian interval distances use constant acceleration within each interval: s = (u + v)t / 2.

M4NA Framework

Force value
fᵢ = kvᵢ

Force quantity
F = Σfᵢtᵢ

Final velocity —

Force values (intervals) —

Force quantity F —

Accumulated distance S —

Average velocity V = S/T —

M4NA force quantities are displayed according to the M4NA definitions. They are not labelled as Newtonian force impulse.

3. Interval-by-interval comparison

Newtonian force values are calculated from the velocity change over each interval. M4NA force values are calculated from the M4NA force–velocity relationship.

Interval	Time (s)	Newton velocity (m/s)	Newton force (N)	Newton distance (m)	M4NA velocity (m/s)	M4NA force value	M4NA distance (m)	M4NA Fᵢ
Select a pattern and calculate to display results.

4. M4NA identity tests

Force–distance identity: F = kS

Calculate to examine this identity.

Interval identity: Fᵢ = ksᵢ

Calculate to examine interval identities.

Force-value and velocity relationship: fᵢ = kvᵢ

Calculate to examine this relationship.

Average relationship: F/T = kV

Calculate to examine this relationship.

The tests check arithmetic consistency using the selected k, interval values and calculated quantities. They are not, by themselves, experimental verification of a physical theory.

5. Velocity-pattern graph

Both series use the selected target velocity pattern. The graph makes the shared velocity values visible; force and distance results are calculated separately using each framework's equations.

6. Cumulative travelling-distance graph

7. How to interpret the comparison

Important: This panel uses the same target interval-end velocity pattern to make the force and distance calculations comparable. Newtonian force is calculated from the velocity change and the entered mass. M4NA force values are calculated from fᵢ = kvᵢ.

In the uniform-velocity example, the Newtonian velocity change is zero, so its calculated resultant force is zero. M4NA gives a nonzero force value for nonzero velocity when k is nonzero. The frameworks' force quantities have different definitions; they should not be treated as identical physical units.

Velocity and Average Velocity: Two Methods of Calculation

Purpose: Compare conventional average velocity calculated from total travelling distance and total time with the arithmetic mean of endpoint instantaneous velocities. Compare both with the M4NA interval-based calculation using the same listed velocity values.

Initial condition: Both examples begin with an initial velocity of 0 m/s. This is the velocity at the starting instant; it does not mean that the object remains motionless throughout the first second.

Input A — Endpoint velocity values

Enter the conventional instantaneous velocity at the end of each interval. These values are also used as the M4NA interval velocity values.

Interval 1 (m/s)

Interval 2 (m/s)

Interval 3 (m/s)

For the conventional calculation, the velocity is assumed to change uniformly within each interval. The initial velocity is 0 m/s.

Input B — Interval durations

Enter the duration of each successive interval.

Interval 1 (s)

Interval 2 (s)

Interval 3 (s)

Calculation 1 — Conventional Framework

For each interval, assuming uniform change of velocity:

Interval travelling distance
sᵢ = (initial interval velocity + final interval velocity) × tᵢ / 2

Quantity	Interval 1	Interval 2	Interval 3	Total

Method A: Average velocity from total distance

3 m/s

Method B: Arithmetic mean of endpoint instantaneous velocities

4 m/s

The arithmetic mean above is the unweighted mean of the three endpoint velocity values. It is not automatically equal to the conventional average velocity calculated from total travelling distance and time.

Calculation 2 — M4NA Framework

M4NA treats each entered velocity value as the velocity associated with the actual travelling distance during that interval.

Interval travelling distance
sᵢ = vᵢtᵢ
Total travelling distance
S = s₁ + s₂ + s₃
Average velocity
V = S / T

Quantity	Interval 1	Interval 2	Interval 3	Total

M4NA average velocity = total travelling distance / total time

4 m/s

Under the M4NA interpretation used here, velocity, average velocity, and instantaneous velocity are assigned the same numerical value for each specified interval. This is M4NA terminology, not the conventional definition of instantaneous velocity.

Results — Identify the Mismatches

Comparison	Calculated result

Calculation assumptions: all listed velocities are nonnegative and the object travels in one direction. Conventional interval distances are calculated by assuming uniform velocity change from the initial velocity of 0 m/s to the first endpoint velocity, then between successive endpoint velocities. M4NA interval distances are calculated directly from the entered interval velocity values and durations.

M4NA Interactive Mechanics Laboratory

Velocity, Travelling Distance & Applied Force

Examine the interval-by-interval calculations and overall results of conventional Newtonian mechanics and the M4NA classical mechanics framework.

1. Operation Inputs

Enter the three interval velocity values and durations. The same inputs are used to display both frameworks, but each framework applies its own stated calculation rules.

Velocity values and interval durations

Interval 1 velocity (m/s)

Interval 2 velocity (m/s)

Interval 3 velocity (m/s)

Interval 1 duration (s)

Interval 2 duration (s)

Interval 3 duration (s)

Force calculation parameters

Newtonian mass (kg)

M4NA locked proportionality constant (k)

2. Conventional Newtonian Mechanics

Distance for each interval under uniform acceleration:
sᵢ = (uᵢ + vᵢ)tᵢ / 2

Interval acceleration and applied force:
aᵢ = (vᵢ − uᵢ) / tᵢ and fᵢ = maᵢ

Average velocity over the complete operation:
V = total travelling distance / total time

For this example, the initial instantaneous velocity is 0 m/s at the starting instant t = 0. It is not treated as the velocity maintained throughout the first interval. Each interval is calculated using its starting and ending instantaneous velocities, assuming uniform acceleration within that interval.

Interval	Start v (m/s)	End v (m/s)	Time (s)	Distance (m)	Acceleration (m/s²)	Applied force value (N)

Total travelling distance 9 m

Total time 3 s

Average velocity 3 m/s

Overall average acceleration 2 m/s²

Overall average applied force value 2 N

Arithmetic mean of the three entered endpoint velocities 4 m/s

Interpretation: Newtonian interval distances are calculated from the starting and ending instantaneous velocities and the assumed motion within each interval. Applied force values are calculated from mass multiplied by interval acceleration. They are not added together to obtain an overall force value.

3. M4NA Classical Mechanics

M4NA interval travelling distance:
sᵢ = vᵢtᵢ

M4NA force-value relationship:
fᵢ = kvᵢ

M4NA force-quantity relationship:
Fᵢ = ksᵢ and F = kS

Overall relationship:
S = Σsᵢ | T = Σtᵢ | V = S/T | F/T = kV

In this M4NA example, the entered velocity value for each interval is used to calculate the travelling distance during that interval. The same locked value of k is used throughout the operation.

Interval	Velocity vᵢ (m/s)	Time tᵢ (s)	Distance sᵢ (m)	Force value fᵢ = kvᵢ	Force quantity Fᵢ = ksᵢ

Total travelling distance S 12 m

Total time T 3 s

Average velocity V = S/T 4 m/s

Overall force quantity F = kS 12k

Force quantity per unit time F/T 4k

Force value corresponding to V: kV 4k

M4NA relationship check: The calculated values will appear here.

4. Side-by-Side Results

Quantity	Newtonian mechanics	M4NA framework
Interval travelling distance	1 m, 3 m, 5 m	2 m, 4 m, 6 m
Total travelling distance	9 m	12 m
Total time	3 s	3 s
Average velocity	3 m/s	4 m/s
Interval applied force values	2 N, 2 N, 2 N	2k, 4k, 6k
Overall applied force value	2 N	4k (F/T = kV)
Force quantity	Not represented here by the M4NA definition F = kS	12k

Purpose of the comparison: The table makes differences in interval-distance calculations, overall travelling distance, average velocity, and force representation visible. The two frameworks' quantities are calculated according to their respective stated definitions; matching labels do not by themselves establish that the underlying concepts or formulas are identical.

M4NA Interactive Mechanics Laboratory

Comparison of Velocity, Travelling Distance and Applied Force: Newtonian Mechanics vs M4NA

Compare interval-by-interval calculations and overall results under conventional Newtonian mechanics and the M4NA classical mechanics framework.

1. Operation Inputs

The same entered velocity values and durations are used in both displays. Each framework applies its own stated calculation rules.

Velocity values and interval durations

Interval 1 velocity (m/s)

Interval 2 velocity (m/s)

Interval 3 velocity (m/s)

Interval 1 duration (s)

Interval 2 duration (s)

Interval 3 duration (s)

Force calculation parameters

Newtonian mass (kg)

M4NA locked proportionality constant (k)

2. Conventional Newtonian Mechanics

Interval travelling distance under uniform acceleration:
sᵢ = (uᵢ + vᵢ)tᵢ / 2

Interval acceleration and applied force value:
aᵢ = (vᵢ − uᵢ) / tᵢ | fᵢ = maᵢ

Average velocity over the complete operation:
V = total travelling distance / total time

The initial instantaneous velocity is 0 m/s at the starting instant t = 0. It is not the velocity maintained throughout the first interval. The calculation assumes uniform acceleration within each interval.

Interval	Start v (m/s)	End v (m/s)	Time (s)	Distance (m)	Acceleration (m/s²)	Applied force value (N)

Total travelling distance 9 m

Total time 3 s

Average velocity 3 m/s

Overall average acceleration 2 m/s²

Overall average applied force value 2 N

Arithmetic mean of entered endpoint velocities 4 m/s

Interpretation: Newtonian interval distances are calculated from the starting and ending instantaneous velocities under the stated uniform-acceleration assumption. The interval force values are calculated using mass multiplied by acceleration; the interval values are not added together to obtain an overall force value.

3. M4NA Classical Mechanics

Interval travelling distance:
sᵢ = vᵢtᵢ

Force-value relationship:
fᵢ = kvᵢ

Force-quantity relationship:
Fᵢ = ksᵢ | F = kS

Overall relationships:
S = Σsᵢ | T = Σtᵢ | V = S/T | F/T = kV

The entered velocity value is used to calculate the actual travelling distance for each M4NA interval. The same locked k is applied throughout the operation.

Interval	Velocity vᵢ (m/s)	Time tᵢ (s)	Distance sᵢ (m)	Force value fᵢ = kvᵢ	Force quantity Fᵢ = ksᵢ

Total travelling distance S 12 m

Total time T 3 s

Average velocity V = S/T 4 m/s

Overall force quantity F = kS 12k

Force quantity per unit time F/T 4k

Force value corresponding to V: kV 4k

M4NA relationship check: F/T and kV will be compared here.

4. Side-by-Side Comparison

Quantity	Newtonian Mechanics	M4NA Framework
Interval travelling distance	1 m, 3 m, 5 m	2 m, 4 m, 6 m
Total travelling distance	9 m	12 m
Total time	3 s	3 s
Average velocity	3 m/s	4 m/s
Interval applied force values	2 N, 2 N, 2 N	2k, 4k, 6k
Overall average applied force value	2 N	4k
Force quantity	Not represented by the M4NA definition F = kS	12k

Purpose of the comparison: Examine how the two frameworks calculate interval travelling distance, total travelling distance, average velocity and applied force. The calculations are presented according to each framework's stated definitions; the comparison itself does not assume that their concepts or formulas are identical.

M4NA Comparative Motion Analysis Lab

An Interval-by-Interval Comparison of Velocity, Travelling Distance, and Applied Force in Newtonian Mechanics and M4NA Classical Mechanics.

1. Velocity, Travelling Distance and Applied Force

Mass, m (kg)

Initial velocity, u (m/s)

Acceleration, a (m/s²)

Locked proportionality constant, k

Interval durations, Δt (s)

M4NA velocity pattern, vᵢ

M4NA force-value input

Force values, fᵢ (when selected)

Enter one duration per interval (for example, 1,1,1,1). Newtonian velocity is evaluated at the end of each successive interval; the time column displays each interval duration, not cumulative elapsed time. The same locked k is used throughout the M4NA operation.

D. Interval-by-Interval Results

Interval	Interval duration, Δt (s)	Newtonian velocity, v (m/s)	Newtonian travelling distance, s (m)	M4NA velocity, vᵢ (m/s)	M4NA travelling distance, sᵢ (m)	M4NA force value, fᵢ	M4NA force quantity, Fᵢ	Newtonian applied/resultant force, ma (N)

E. Overall Results

Final Newtonian velocity8 m/s

Newtonian travelling distance at final time16 m

M4NA travelling distance, S = Σsᵢ60 m

M4NA force quantity, F = ΣFᵢ60

Final elapsed time4 s

M4NA average velocity, V = S/T15 m/s

Newtonian force, F = ma2 N

Locked k1

The M4NA force-value pattern is calculated from the selected input mode and the same locked k. For each interval, the displayed M4NA force quantity uses Fᵢ = fᵢΔtᵢ and travelling distance uses sᵢ = vᵢΔtᵢ. The duration column shows interval durations; elapsed time accumulates across intervals for the Newtonian calculations.

2. Newtonian Travelling-Distance Equation Comparison

A. Newtonian equation

s = ut + ½at²

B. Average-velocity form

s = ((u + v) / 2)t

C. Final-velocity form

s = vt

Results for the same Newtonian inputs and time entries

Interval	Interval duration, Δt (s)	Newtonian velocity at interval end, v = u + aτ (m/s)	A. s = ut + ½at² (m)	B. s = ((u + v)/2)t (m)	C. s = vt (m)
Sum of interval values	—	—	16	16	20

Equations A and B calculate the travelling distance within each interval using the velocity at the interval start and the interval duration. Equation C uses the velocity at the interval end multiplied by that same interval duration. The elapsed time τ accumulates across intervals even though the duration column remains 1, 1, 1, 1 s.