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.
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.
2Average Velocity
Calculate average velocity from total distance and total elapsed time, or calculate a missing quantity.
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.
| Interval | Force value fᵢ | Velocity vᵢ (m/s) | Duration tᵢ (s) | Distance sᵢ (m) | Force quantity Fᵢ | Action |
|---|
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.
5Force–Travel Proportionality
Calculate one missing quantity in the relationship F = kS.
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.
Part V — Applied-Force Identity
Reconstruct, examine and compare force–velocity–travel patterns using the same three identity relationships.
7Applied Force Pattern Reconstruction
Enter force values fᵢ, interval durations tᵢ and a fixed k. The panel reconstructs vᵢ = fᵢ/k, then checks all three relationships interval by interval and for the total.
8Force–Velocity Pattern Check
Use the same input pattern structure: force values, velocity values and durations. Check fᵢ ∝ vᵢ and fᵢtᵢ ∝ vᵢtᵢ in each interval, plus the corresponding total relationships.
9Pattern Identity Test
Enter three patterns using the same force-value, velocity and duration inputs. The test applies the same three identity relationships to every interval and to the total: Fᵢ = ksᵢ, Fᵢ/tᵢ = k(sᵢ/tᵢ), fᵢ = kvᵢ; and F = kS, F/T = k(S/T), F/T = kV. It then compares the three patterns.
10Resultant Force Calculator
Enter two or three forward patterns and two or three opposing restrictive patterns. The calculator generates the resultant force-value pattern, then checks the three applied/resultant force identity relationships for every interval and for the total.
Forward force patterns
Opposing restrictive force patterns
Resultant velocity values are derived using the fixed k.
11Applied/Resultant Force Identity Test
Enter three patterns with force values, velocity values and interval durations. The test applies the same three identity relationships to every interval and to the total, including the total-level force-value/velocity relationship F/T = kV.
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
2. Moving Observer
The observer travels with the train and measures the ball relative to the train.
s = 5 × 4 = 20 m
3. Static Observer
The observer remains on the ground and measures the train and ball relative to the ground.
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 |
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.
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.
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
2. Results at a glance
Newtonian calculation
M4NA calculation
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.
5. Velocity pattern and accumulated distance
Compare the entered velocity pattern and the cumulative distance generated by each framework.
6. Interpretation
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.
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
Force calculation parameters
2. Conventional Newtonian Mechanics
sᵢ = (uᵢ + vᵢ)tᵢ / 2
aᵢ = (vᵢ − uᵢ) / tᵢ and fᵢ = maᵢ
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) |
|---|
3. M4NA Classical Mechanics
sᵢ = vᵢtᵢ
fᵢ = kvᵢ
Fᵢ = ksᵢ and F = kS
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 |
|---|
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 |
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
Force calculation parameters
2. Conventional Newtonian Mechanics
sᵢ = (uᵢ + vᵢ)tᵢ / 2
aᵢ = (vᵢ − uᵢ) / tᵢ | fᵢ = maᵢ
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) |
|---|
3. M4NA Classical Mechanics
sᵢ = vᵢtᵢ
fᵢ = kvᵢ
Fᵢ = ksᵢ | F = kS
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 |
|---|
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 |
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
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
2. Newtonian Travelling-Distance Equation Comparison
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 |
Mechanics for New Age · Interactive Research Lab
Force Value, Force Quantity & Travelling Distance
Compare the Newtonian calculations with the M4NA relationships using an adjustable example. Interactive controls are available to signed-in users.
Framework equations
Newtonian comparison
M4NA comparison
Comparison convention: for each interval, the M4NA assigned velocity is the Newtonian endpoint velocity. The two interval-distance calculations are displayed separately.
Set the example
Preset: u = 0, a = 2 m/s², m = 1 kg, four intervals of 1 second, k = 1.
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You can view the formulas and preset results. Sign in to change inputs, recalculate, or restore the preset.
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Calculated overview
Force value and force quantity by interval
| Interval | Duration (s) | Newtonian endpoint velocity (m/s) | Newtonian resultant force, ma | M4NA force value, fᵢ | M4NA force quantity, Fᵢ |
|---|---|---|---|---|---|
| 1 | 1 | 2 | 2 | 2 | 2 |
| 2 | 1 | 4 | 2 | 4 | 4 |
| 3 | 1 | 6 | 2 | 6 | 6 |
| 4 | 1 | 8 | 2 | 8 | 8 |
| Summary | 2 each interval | See interval values | 20 | ||
The force quantity summary is the sum of the interval force quantities. The force values themselves remain listed by interval.
Travelling-distance comparison by interval
| Interval | Starting velocity (m/s) | Endpoint velocity (m/s) | Newtonian interval distance (m) | M4NA assigned velocity (m/s) | M4NA interval distance, sᵢ (m) |
|---|---|---|---|---|---|
| 1 | 0 | 2 | 1 | 2 | 2 |
| 2 | 2 | 4 | 3 | 4 | 4 |
| 3 | 4 | 6 | 5 | 6 | 6 |
| 4 | 6 | 8 | 7 | 8 | 8 |
| Total travelling distance | 16 | — | 20 | ||
M4NA identity checks
Endpoint velocity × duration
In the preset example, the first expression gives 2, 4, 6, and 8 m; the Newtonian interval calculation gives 1, 3, 5, and 7 m.
M4NA · Interactive Mechanics Laboratory · Force value, force quantity, and travelling-distance comparison.
M4NA Interactive Mechanics Laboratory
Static Observer vs Moving Observer
Explore the train-and-ball operation from two reference views.
Change the operating conditions, calculate the results, and run the animation.
1. Operation Controls
2. Synchronized Observer Views
Train and ball relative to ground
The train moves through the ground reference. The red marker indicates the ball's calculated ground position.
Ball relative to the train
The train remains fixed in this view while the ball moves forward or backward relative to it.
3. Moving Observer — Ball Relative to Train
4. Static Observer — Ball Relative to Ground
5. Comparison of Both Observations
| Quantity | Moving observer | Static observer |
|---|---|---|
| Reference | Train | Ground |
| Train velocity | Train is fixed in this view | 20 m/s |
| Ball relative velocity | 5 m/s forward | Combined with train velocity |
| Ball ground velocity | Train-relative value is shown separately | 25 m/s |
| Ball travelling distance | 50 m relative to train | 250 m relative to ground |
| Operating time | 10 s | 10 s |
6. Travelling Distance Comparison
Compare the travelling-distance magnitudes for the selected operation.
7. Position Over Time
Compare train ground position, ball ground position, and ball position relative to the train.
8. Observation Interpretation
M4NA Interactive Mechanics Laboratory
Static Observer vs Moving Observer
Explore the motion of a train and an object inside it from two observer perspectives,
together with travelling-distance and position–time graphs.
You can view this laboratory, but you must sign in to operate its calculators and interactive animations.
Sign In Register1. Experiment Settings
Set the experiment parameters once. All four animations use these same inputs, but each animation runs independently.
2. Moving Observer — Inside the Train
The observer travels with the train and observes the object's movement relative to the train.
3. Static Observer — On the Ground
The observer remains on the ground and observes the train and object relative to the ground reference.
4. Travelling-Distance Graph
Watch the travelling-distance values develop over time. This graph has its own playback clock.
5. Position–Time Graph
Compare the train's ground-frame position with the object's ground-frame position. Its animation runs independently of the other three.
How Independent Playback Works
- Each component has its own Play, Pause, and Reset buttons.
- Pausing one component does not pause the other three.
- Resetting one component does not reset the other three.
- Selecting Calculate / Apply Settings to All stops and resets all four components using the new settings.
- The playback-speed slider changes the animation rate, not the calculated velocities or distances.
Model used in this visualisation: train motion is represented along a positive ground direction. The object's signed ground-frame velocity is calculated as train velocity plus its signed velocity relative to the train. The graphs show modelled positions and travelling distances for the selected experiment.
M4NA Interactive Mechanics Laboratory
Static Observer vs Moving Observer
Explore the train, the moving ball, and the two observer perspectives
through synchronized animation, calculations, and position–time graphs.
Experiment Configuration
Calculated Results
Ball ground velocity = train velocity + direction sign × ball velocity relative to train.
Live Experiment Readout
Two Observer Perspectives
The drawings are illustrative. Numerical readouts and graphs show the positions calculated from the configured velocities and elapsed time.
Moving Observer — Inside the Train
The train is the observer's reference frame.
Static Observer — On the Ground
The ground is the observer's reference frame.
Position and Travelling-Distance Summary
| Quantity | At start | At current time | At final time |
|---|---|---|---|
| Elapsed time (s) | 0.00 | 0.00 | — |
| Train travelling distance (m) | 0.00 | 0.00 | — |
| Ball travelling distance relative to train (m) | 0.00 | 0.00 | — |
| Ball travelling distance relative to ground (m) | 0.00 | 0.00 | — |
| Train position (m) | — | — | — |
| Ball ground position (m) | — | — | — |
Progressive Travelling-Distance Graph
The bars grow as the experiment progresses. During playback, each bar displays the distance reached at the current elapsed time rather than immediately displaying its final length.
Progressive Position–Time Graph
The curves are drawn progressively up to the current elapsed time. The vertical indicator and the position markers move with the animation.
Experiment Interpretation
Train position = initial train position + train velocity × elapsed time.
Ball position in train = initial ball position + relative velocity × elapsed time.
Ball ground position = train position + ball position relative to train.
This widget uses a constant-velocity model for the configured experiment. The selected direction determines whether the ball's velocity relative to the train is in the same or opposite direction as the train. These equations describe the model used by this demonstration.