Static hold
Zero velocity with a static-hold label means the required friction is within capacity, not that mu_s x N is fully mobilized.
Physics calculator
Friction Trajectory Calculator
Model a one-dimensional friction trajectory with a static-friction start gate, kinetic-friction motion, exact zero-speed events, and live position and velocity history.
Piecewise motion under dry friction
Will the object stay at rest, stop within the interval, reverse, or still be sliding at the requested horizon?
Follow an object along a signed axis on an incline. The model first decides whether static friction can hold, then applies kinetic friction during each sliding segment and rechecks the static gate whenever velocity reaches zero.
Designed for: For students, lab teams, and preliminary design checks that need the full time history rather than a single stopping-distance number.
Position at duration -
Velocity at duration -
First stop time -
Position at first stop -
Normal force -
Final motion state -
LIVE MODEL OUTPUT
Current trajectory samples
Time-ordered state produced from the current inputs and every zero-speed event.
| Time (s) | Friction state | Position (m) | Velocity (m/s) | Acceleration (m/s^2) |
|---|
CURRENT CALCULATION PROCESS
Formula, units, substitution, intermediate quantities, and check
N = m x g x cos(theta); D = F - m x g x sin(theta); |D| <= mu_s x N holds at rest; a = [D - sign(v) x mu_k x N] / m while sliding
The axis is fixed to the plane. Motion is integrated in constant-acceleration pieces. If a piece would cross zero velocity, the exact crossing time is solved first; the remaining time is then evaluated under the static-friction rule before any reverse motion begins.
| Symbol | Meaning | Unit | Default |
|---|---|---|---|
| m | Translating mass | kg | 12 |
| v0 | Initial signed velocity | m/s | 8 |
| mu_s | Static friction coefficient | unitless | 0.42 |
| mu_k | Kinetic friction coefficient | unitless | 0.30 |
| theta | Incline angle, positive upslope | deg | 0 |
| F | Signed applied force along plane | N | 0 |
| T | Modeled duration | s | 5 |
| dt | Output sample interval | s | 0.25 |
Conversions and rounding: Angles are converted from degrees to radians for sin and cos. All other defaults are already SI. Display values are rounded, but event times and integration retain full floating-point precision.
HOW TO USE
Build a defensible friction trajectory
- Choose a signed axis and enter velocity and force with that same direction convention.
- Enter static and kinetic coefficients from the same contact pair and surface condition.
- Set the incline angle; use zero for a horizontal surface and keep applied force parallel to the plane.
- Choose a duration long enough to include the possible stop and a sample interval fine enough to inspect it.
- Read the state table and chart together, then use the first-stop result only if the model actually reaches zero velocity.
SUBJECT FUNDAMENTALS
What determines the trajectory
- Static friction is a response
- At rest, friction takes the magnitude needed for equilibrium up to mu_s x N; it is not automatically equal to its maximum.
- Kinetic friction has a direction
- During sliding, mu_k x N acts opposite the instantaneous velocity, so its sign can change after a reversal.
- The incline changes two terms
- Gravity contributes m x g x sin(theta) along the plane and leaves m x g x cos(theta) to create normal force.
- Stopping is an event
- A time step must not jump through zero velocity. The model solves the crossing exactly and then tests static friction again.
- Position can remain nonzero
- A stopped object holds its accumulated displacement; velocity goes to zero but the earlier travel is not erased.
- Coefficients describe conditions
- Dry, clean, contaminated, hot, or polished interfaces can have materially different coefficients even for the same materials.
RESULT INTERPRETATION
Read the path as a sequence of regimes
Sliding label
Upslope or downslope describes velocity direction. Acceleration may point the other way while the object slows.
First stop
The first stop is a diagnostic event. Later motion can occur if the static gate fails after that event.
DEEPER ANALYSIS
Three trajectory decisions hidden by one-line formulas
Stopping versus reversing
If the non-friction drive is inside the static limit when speed reaches zero, the object stays there. If it exceeds the limit, a new kinetic segment starts in the drive direction.
Sampling versus event precision
The table is sampled at dt, but stop time is not rounded to that grid. This prevents a coarse reporting interval from moving the physical stop point.
Changing normal load
This page assumes the applied force is parallel to the plane. A rope angled away from the surface would reduce N and both friction limits; use a force-component model instead.
WORKED CASES
Worked motion cases
Warehouse tote stopping on level flooring
For m = 12 kg, v0 = 8 m/s, mu_k = 0.30, and no applied force, acceleration is about -2.942 m/s^2. The tote stops near 2.72 s and 10.88 m, then static friction holds because the horizontal drive is zero.
Cart released on a steep ramp
With v0 = 0, the model first compares m x g x sin(theta) with mu_s x m x g x cos(theta). If tan(theta) exceeds mu_s, the cart starts downslope; entering a small nonzero velocity is not required to trigger motion.
ASSUMPTIONS
Model assumptions to record
- One-dimensional translation along a rigid plane.
- Constant mass, incline, gravity, applied force, and friction coefficients.
- No aerodynamic drag, rolling resistance, wheel inertia, deformation, or speed-dependent friction.
- Continuous contact with no applied force component normal to the plane.
- mu_s >= mu_k so the launch rule is mechanically coherent for this simplified dry-friction model.
TECHNICAL LANGUAGE
Trajectory-specific terms
- Static limit
- The largest friction magnitude available while the contact has no relative sliding.
- Kinetic segment
- A time interval with nonzero sliding velocity and a fixed friction direction.
- Drive force
- The signed sum of applied and along-plane gravity forces before friction is included.
- Zero-speed event
- The exact instant a sliding solution reaches v = 0 and must be reclassified.
- Stopping distance
- Position accumulated from the starting point to the first zero-speed event.
- Signed axis
- A declared positive direction that gives force, velocity, acceleration, and position consistent signs.
EVIDENCE AND DATA LINEAGE
Evidence to preserve with a result
Record the contact materials and condition, how mu_s and mu_k were obtained, incline measurement, force direction, mass, time horizon, sample interval, and the exported table. A coefficient copied from a generic table is an assumption, not a test result for the actual interface.
LIMITS AND EXCLUSIONS
Where this trajectory should not be used
- Do not use for rolling tires, bearings, lubricated contacts, or stick-slip systems without a validated friction law.
- Do not use after contact loss, impacts, or geometry changes.
- The model does not include speed-dependent drag or a force that varies with time or position.
- A result near a friction threshold is sensitive to coefficient uncertainty and surface contamination.
- This is a deterministic nominal path, not a confidence interval or safety certification.
RELIABLE SOURCES
Primary references and stated use
- OpenStax University Physics, 6.2 Friction - Static and kinetic friction laws and their empirical boundary.
- OpenStax University Physics, 3.4 Motion with Constant Acceleration - Piecewise kinematic equations used between events.
- NIST SI base and derived units - SI units for mass, time, force, velocity, and acceleration.
FREQUENTLY ASKED QUESTIONS
Questions about friction paths and stopping events
Why can the stop time fall between table rows?
The solver locates the exact zero of v within a constant-acceleration segment. The sample interval controls reporting density, not event accuracy.
Why is static friction checked after a stop?
Kinetic friction no longer applies once sliding velocity is zero. The object can remain stopped only if static friction can balance the drive.
Can I enter a negative incline?
Yes. A negative angle makes gravity act toward the positive axis under the stated sign convention.
Why must mu_s be at least mu_k here?
The simplified launch-and-slide rule assumes a larger or equal breakaway capacity. If your measured interface violates that relation, it needs a more specific friction model.
Does a zero final velocity prove the object never moved?
No. Check end position and the trajectory table; the object may have traveled and then stopped.
Can I model a force pulse?
Not on this page. Applied force is constant. Split the motion into separately documented intervals or use a time-varying dynamics solver.
IMPORTANT NOTE
Engineering and laboratory use note
This calculator implements an idealized Coulomb-friction model for education and preliminary analysis. Verify coefficients and forces experimentally before relying on the result for safety, braking, lifting, transport, or equipment design.