Account for mass crossing a momentum boundary
The identity F = ma assumes a particle of fixed mass. A subsystem that loses mass needs an additional momentum-flux account. Consider an abstract one-dimensional retained body with no external force, initially at rest. An infinitesimal positive mass dμ leaves with velocity v − u relative to the inertial frame, where u is its fixed backward speed relative to the body.
Learning outcome: Derive the differential boundary balance before integrating it. Compare two initial masses at equal fractions, and then compare constant thrust with increasing acceleration.
Starting recap
Integrate signed force and normalized mass moments. Specify the system before conserving momentum, and independently test energy and observation constraints.
All lessons are open directly. Your saved experiments remain when you leave and return.
Try
Change the retained fraction to 0.8 at any supported initial mass. The exact change is 100 ln(1.25) ≈ 22.3144 m/s, whereas the fixed-initial-mass shortcut gives 20 m/s.
Open the experimentPredict
At final retained fraction 0.6 and u = 100 m/s, the ideal velocity change is:
Experiment
Derive the differential boundary balance before integrating it. Compare two initial masses at equal fractions, and then compare constant thrust with increasing acceleration.
Keep expelled mass in the momentum boundary
Worked teaching model · synthetic values · Pure mathematical one-dimensional mass-expulsion model in an inertial force-free environment. Relative expelled speed100 m/s, initial rest and constant mass-loss rate over10 s. No fuel, device, pressure, design, trajectory targeting or physical propulsion instructions. Total retained-plus-expelled mass is closed; retained subsystem is not.
Reset keeps your prediction, captures and practice history.
Retained speed from the changing-mass balance
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: 100 ln(m0/m(t)) (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Retained speed from the changing-mass balance
100 ln(m0/m(t))
| elapsed time (s) | retained speed (m/s) |
|---|---|
| 0 | 0 |
| 0.125 | 0.5012542 |
| 0.25 | 1.005034 |
| 0.375 | 1.511364 |
| 0.5 | 2.020271 |
| 0.625 | 2.531781 |
| 0.75 | 3.045921 |
| 0.875 | 3.562718 |
| 1 | 4.082199 |
| 1.125 | 4.604394 |
| 1.25 | 5.129329 |
| 1.375 | 5.657035 |
| 1.5 | 6.18754 |
| 1.625 | 6.720875 |
| 1.75 | 7.257069 |
| 1.875 | 7.796154 |
| 2 | 8.338161 |
| 2.125 | 8.883121 |
| 2.25 | 9.431068 |
| 2.375 | 9.982034 |
| 2.5 | 10.53605 |
| 2.625 | 11.09316 |
| 2.75 | 11.65338 |
| 2.875 | 12.21676 |
| 3 | 12.78334 |
| 3.125 | 13.35314 |
| 3.25 | 13.92621 |
| 3.375 | 14.50258 |
| 3.5 | 15.08229 |
| 3.625 | 15.66538 |
| 3.75 | 16.25189 |
| 3.875 | 16.84187 |
| 4 | 17.43534 |
| 4.125 | 18.03236 |
| 4.25 | 18.63296 |
| 4.375 | 19.23719 |
| 4.5 | 19.84509 |
| 4.625 | 20.45672 |
| 4.75 | 21.0721 |
| 4.875 | 21.6913 |
| 5 | 22.31436 |
| 5.125 | 22.94132 |
| 5.25 | 23.57223 |
| 5.375 | 24.20716 |
| 5.5 | 24.84614 |
| 5.625 | 25.48922 |
| 5.75 | 26.13648 |
| 5.875 | 26.78794 |
| 6 | 27.44368 |
| 6.125 | 28.10375 |
| 6.25 | 28.76821 |
| 6.375 | 29.43711 |
| 6.5 | 30.11051 |
| 6.625 | 30.78848 |
| 6.75 | 31.47107 |
| 6.875 | 32.15836 |
| 7 | 32.85041 |
| 7.125 | 33.54727 |
| 7.25 | 34.24903 |
| 7.375 | 34.95575 |
| 7.5 | 35.66749 |
| 7.625 | 36.38434 |
| 7.75 | 37.10637 |
| 7.875 | 37.83364 |
| 8 | 38.56625 |
| 8.125 | 39.30426 |
| 8.25 | 40.04776 |
| 8.375 | 40.79682 |
| 8.5 | 41.55154 |
| 8.625 | 42.312 |
| 8.75 | 43.07829 |
| 8.875 | 43.8505 |
| 9 | 44.62871 |
| 9.125 | 45.41303 |
| 9.25 | 46.20355 |
| 9.375 | 47.00036 |
| 9.5 | 47.80358 |
| 9.625 | 48.6133 |
| 9.75 | 49.42963 |
| 9.875 | 50.25268 |
| 10 | 51.08256 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Final retained mass | 9 | kg |
| Expelled total mass | 6 | kg |
| Constant mass-loss magnitude | 0.6 | kg/s |
| Momentum-flux thrust magnitude | 60 | N |
| Final retained velocity change | 51.08256 | m/s |
| Final retained acceleration | 6.666667 | m/s² |
| Fixed-initial-mass shortcut velocity change | 40 | m/s |
The expelled mass carries momentum across the retained subsystem boundary; its contribution cannot be omitted.
For positive expelled amount dmu=−dm, momentum balance gives m dv=ve dmu=−ve dm to first order. Integrate dv=−ve dm/m to obtain ve ln(m0/mf). Constant mass-loss rate gives constant momentum-flux thrust but increasing acceleration as retained mass falls. This is not d(mv)/dt=0 for the retained subsystem and not a fuel or hardware model.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The expelled mass carries momentum across the retained subsystem boundary; its contribution cannot be omitted.
For positive expelled amount dmu=−dm, momentum balance gives m dv=ve dmu=−ve dm to first order. Integrate dv=−ve dm/m to obtain ve ln(m0/mf). Constant mass-loss rate gives constant momentum-flux thrust but increasing acceleration as retained mass falls. This is not d(mv)/dt=0 for the retained subsystem and not a fuel or hardware model.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Which matter must be included for the zero-external-force momentum conservation statement?
Continue
Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.
Skipping retains experiments and awards no mastery.
Save a science backup or import one
Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.
A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.
Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.
Complete file sets and course backups
A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.
The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.
0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.
Science storage: loading. No account or remote computation is required.