Two-axis thruster maneuver
A test engineer applies one translation pulse and then a trim pulse, checking final momentum components before converting to velocity with vehicle mass.
Physics and mechanics
Momentum Trajectory Calculator
Trace px and py through two constant-force phases, report vector impulse and final momentum, and plot the path in momentum space rather than physical position.
TWO-DIMENSIONAL MOMENTUM PATH
Trace a vector through momentum space under two force phases
This calculator serves dynamics students and preliminary control or propulsion studies that know initial momentum and two constant-force phases. It updates x and y components independently, then plots the ordered path in the px-py plane. The curve is a momentum trajectory—not the object’s spatial trajectory—and it omits mass-dependent position and velocity integration.
Final px (kg*m/s)-
Final py (kg*m/s)-
Final momentum magnitude-
Final vector direction-
Total x impulse (N*s)-
Total y impulse (N*s)-
Elapsed time (s)-
TWO-DIMENSIONAL MOMENTUM PATH
Two-phase vector momentum ledger
The plotted path shows how the momentum vector changes under the entered force components. It does not show physical displacement, orbit, projectile range, or attitude.
| State | Time (s) | px | py | |p| | Applied impulse |
|---|
DETAILED CALCULATION PROCESS
Formula, units, default substitution, and reconciliation
1. Governing relation
p1 = p0 + F1 Delta t1; p2 = p1 + F2 Delta t2; |p2| = sqrt(px2^2 + py2^2); theta = atan2(py2, px2)
Integrate each constant force component over its phase duration, add the resulting vector impulse to the prior momentum, and calculate magnitude and direction only after both components are known.
2. Symbols and default basis
| Symbol | Meaning | Unit | Default basis |
|---|---|---|---|
| p0 | Initial two-dimensional momentum vector | kg*m/s | (6, 2) |
| F1 | Net force vector during phase 1 | N | (4, 3) |
| Delta t1 | Duration of phase 1 | s | 1.5 |
| F2 | Net force vector during phase 2 | N | (-2, 1) |
| Delta t2 | Duration of phase 2 | s | 2 |
| J | Force-time vector impulse | N*s | F Delta t |
| theta | Direction from +x using atan2 | degrees | derived |
3. Unit and sign normalization
- Each N*s impulse component is numerically equivalent to kg*m/s.
- x and y components are added independently before magnitude is calculated.
- atan2 preserves the quadrant; direction is undefined only when both final components are exactly zero.
4. Current numerical substitution
5. Independent reconciliation
HOW TO USE THIS CALCULATOR
Construct a momentum-space path
- Declare fixed x and y axes before entering any signed component.
- Enter initial momentum components from mass and velocity or another validated state estimate.
- Describe each phase with one constant net-force vector and a positive duration.
- Read the ledger in time order and compare the vector impulse with component changes.
- Use the px-py plot for direction changes; use a separate kinematic model if position, range, or velocity history is required.
MOMENTUM FOUNDATIONS
Five vector-trajectory concepts
- Momentum space
- Coordinate plane whose axes are momentum components, not physical location.
- Vector impulse
- Time integral of force in each direction.
- Component independence
- Orthogonal components satisfy their own impulse-momentum equations.
- Phase switch
- Instant when the declared constant-force vector changes.
- Quadrant-aware direction
- atan2 distinguishes vectors with identical slopes in different quadrants.
DEEP ANALYSIS 1
Mass is absent because momentum is the state
Mass is needed to turn momentum into velocity, but force directly changes momentum through impulse.
DEEP ANALYSIS 2
Straight segments follow constant force
In momentum space, dp/dt = F. A constant force vector therefore traces a straight line.
DEEP ANALYSIS 3
Spatial curvature is a different calculation
A momentum path does not determine position without mass, initial position, and integration of velocity over time.
RESULT INTERPRETATION
Reading direction changes and zero momentum
A final component may be negative even when magnitude is positive; the sign identifies direction along that axis. The angle is measured counterclockwise from positive x and is reported only when the magnitude is nonzero.
A kink at the phase boundary means the force direction changed. It is not an instantaneous momentum jump: momentum remains continuous, while its rate of change switches.
REAL USE CASES
Two momentum-path applications
Deflected particle model
A classroom particle receives a transverse force after an initial push; the plot shows vector redirection without pretending to be its laboratory position track.
EVIDENCE AND DATA QUALITY
Record vectors and timing together
Retain coordinate definitions, initial-state derivation, force-source or sensor records, phase start and end timestamps, net-force construction, omitted forces, units, sign convention, mass if later converting to velocity, and all unrounded component states.
LIMITS AND EXCLUSIONS
What the trajectory leaves out
- Forces are constant within each phase and change instantaneously at the boundary.
- The model tracks momentum, not position, velocity, energy, angular momentum, or attitude.
- Relativistic momentum, variable mass, drag, gravity variation, and uncertainty are excluded.
- A real force waveform requires numerical integration of measured samples rather than two averages.
TERMS USED HERE
Vector momentum terms
- Momentum component
- Projection of momentum on a chosen coordinate axis.
- Vector magnitude
- Square root of summed squared components.
- Impulse vector
- Force-time accumulation with direction.
- Momentum space
- Abstract component space used to represent momentum states.
- Phase
- Time interval governed by one force vector.
- atan2
- Quadrant-aware angle function using both y and x components.
RELIABLE SOURCES
References supporting this model
FREQUENTLY ASKED QUESTIONS
Questions about momentum trajectories
Is this the object’s path through space?
No. The axes are momentum components. Position requires a separate integration of velocity and therefore mass and initial position.
Why are the segments straight?
Each phase uses a constant force vector, so momentum changes linearly with time in both components.
Can final direction be undefined?
Yes. When final px and py are both zero, a zero vector has no direction.
Does a negative component mean negative magnitude?
No. Components are signed; vector magnitude is always nonnegative.
Can I model gravity as one force component?
Yes for a constant-mass, constant-gravity interval if the entered Fy is the net force including gravity and other y-forces.
How do I use a changing force?
Integrate the sampled force components over time with an appropriate numerical method and uncertainty analysis.
IMPORTANT BOUNDARY
Momentum space is not physical space
This two-phase vector model is educational and preliminary. It must not be used alone for navigation, flight, weapons, crash reconstruction, control certification, or structural-load decisions.