Integrate mechanical loss once within a declared system
A2 kg point in a stationary medium experiences linear resistance with coefficient1 kg/s and no drive. From initial velocity v0, the solution is v=v0 exp(−t/2). Its kinetic energy is v0² exp(−t) joules because m/2=1 in this packet.
Learning outcome: Integrate drag power, compare with endpoint kinetic change and write both point-only and point-plus-medium balances without double counting.
Starting recap
Use vector dot products for work, gradients for conservative force, definite integrals for accumulation and explicit energy/frame boundaries. Check accessible states before assigning trajectories.
All lessons are open directly. Your saved experiments remain when you leave and return.
Try
For v(0) = 2 and T = 1, remaining kinetic energy4/e≈1.471518 J and internal gain4(1−1/e)≈2.528482 J. Their sum remains4 J.
Open the experimentPredict
Including the medium in the stipulated closed boundary means:
Experiment
Integrate drag power, compare with endpoint kinetic change and write both point-only and point-plus-medium balances without double counting.
Integrate dissipation into a larger energy account
Worked teaching model · synthetic values · Horizontal point m2 kg with drag coefficient1 kg/s starts at selected positive velocity in a stationary medium, no other forces. v=v0 exp(−t/2). All mechanical loss becomes internal/thermal energy of the selected point-plus-medium boundary by stipulation; no measured heating distribution.
Reset keeps your prediction, captures and practice history.
Mechanical and internal conversion within a chosen boundary
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: mechanical kinetic energy (solid).2: internal/thermal gain (dashed). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Mechanical and internal conversion within a chosen boundary
mechanical kinetic energy
| time (s) | energy (J) |
|---|---|
| 0 | 16 |
| 0.04 | 15.37263 |
| 0.08 | 14.76986 |
| 0.12 | 14.19073 |
| 0.16 | 13.6343 |
| 0.2 | 13.09969 |
| 0.24 | 12.58605 |
| 0.28 | 12.09254 |
| 0.32 | 11.61838 |
| 0.36 | 11.16282 |
| 0.4 | 10.72512 |
| 0.44 | 10.30458 |
| 0.48 | 9.900534 |
| 0.52 | 9.512329 |
| 0.56 | 9.139345 |
| 0.6 | 8.780986 |
| 0.64 | 8.436679 |
| 0.68 | 8.105872 |
| 0.72 | 7.788036 |
| 0.76 | 7.482663 |
| 0.8 | 7.189263 |
| 0.84 | 6.907368 |
| 0.88 | 6.636527 |
| 0.92 | 6.376305 |
| 0.96 | 6.126286 |
| 1 | 5.886071 |
| 1.04 | 5.655275 |
| 1.08 | 5.433528 |
| 1.12 | 5.220477 |
| 1.16 | 5.015779 |
| 1.2 | 4.819107 |
| 1.24 | 4.630147 |
| 1.28 | 4.448597 |
| 1.32 | 4.274165 |
| 1.36 | 4.106572 |
| 1.4 | 3.945551 |
| 1.44 | 3.790844 |
| 1.48 | 3.642203 |
| 1.52 | 3.49939 |
| 1.56 | 3.362177 |
| 1.6 | 3.230344 |
| 1.64 | 3.103681 |
| 1.68 | 2.981984 |
| 1.72 | 2.865058 |
| 1.76 | 2.752718 |
| 1.8 | 2.644782 |
| 1.84 | 2.541079 |
| 1.88 | 2.441442 |
| 1.92 | 2.345711 |
| 1.96 | 2.253735 |
| 2 | 2.165365 |
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.
internal/thermal gain
| time (s) | energy (J) |
|---|---|
| 0 | 0 |
| 0.04 | 0.627369 |
| 0.08 | 1.230138 |
| 0.12 | 1.809273 |
| 0.16 | 2.365699 |
| 0.2 | 2.900308 |
| 0.24 | 3.413954 |
| 0.28 | 3.90746 |
| 0.32 | 4.381615 |
| 0.36 | 4.837179 |
| 0.4 | 5.274879 |
| 0.44 | 5.695417 |
| 0.48 | 6.099466 |
| 0.52 | 6.487671 |
| 0.56 | 6.860655 |
| 0.6 | 7.219014 |
| 0.64 | 7.563321 |
| 0.68 | 7.894128 |
| 0.72 | 8.211964 |
| 0.76 | 8.517337 |
| 0.8 | 8.810737 |
| 0.84 | 9.092632 |
| 0.88 | 9.363473 |
| 0.92 | 9.623695 |
| 0.96 | 9.873714 |
| 1 | 10.11393 |
| 1.04 | 10.34473 |
| 1.08 | 10.56647 |
| 1.12 | 10.77952 |
| 1.16 | 10.98422 |
| 1.2 | 11.18089 |
| 1.24 | 11.36985 |
| 1.28 | 11.5514 |
| 1.32 | 11.72584 |
| 1.36 | 11.89343 |
| 1.4 | 12.05445 |
| 1.44 | 12.20916 |
| 1.48 | 12.3578 |
| 1.52 | 12.50061 |
| 1.56 | 12.63782 |
| 1.6 | 12.76966 |
| 1.64 | 12.89632 |
| 1.68 | 13.01802 |
| 1.72 | 13.13494 |
| 1.76 | 13.24728 |
| 1.8 | 13.35522 |
| 1.84 | 13.45892 |
| 1.88 | 13.55856 |
| 1.92 | 13.65429 |
| 1.96 | 13.74627 |
| 2 | 13.83464 |
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 velocity | 1.471518 | m/s |
| Initial mechanical energy | 16 | J |
| Final mechanical energy | 2.165365 | J |
| Integrated drag work | -13.83464 | J |
| Point-plus-medium internal energy gain | 13.83464 | J |
| Mechanical plus internal total | 16 | J |
The point loses mechanical energy while a larger stipulated boundary retains the corresponding internal energy.
Drag power is−bdrag v²=−v0² exp(−t) W with the supplied m2,bdrag1. Integrating gives work−v0²(1−exp(−T)), equal to Kfinal−Kinitial. Count that loss once: point-only external work and point-plus-medium thermal conversion are two descriptions with different boundaries, not two independent losses.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The point loses mechanical energy while a larger stipulated boundary retains the corresponding internal energy.
Drag power is−bdrag v²=−v0² exp(−t) W with the supplied m2,bdrag1. Integrating gives work−v0²(1−exp(−T)), equal to Kfinal−Kinitial. Count that loss once: point-only external work and point-plus-medium thermal conversion are two descriptions with different boundaries, not two independent losses.
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.
What evidence is missing before converting the internal gain into a predicted temperature rise?
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.