Use v dv/dx without crossing an inaccessible endpoint
For a monotonic piece of one-dimensional motion, the chain rule gives dv/dt=(dv/dx)(dx/dt)=v dv/dx. This can turn a position-dependent force equation into a relation between speed and position, but the selected motion branch must remain explicit.
Learning outcome: Derive the spatial relation from the time equation, test an exact turn and an inaccessible target, and explain where division by velocity would fail.
Starting recap
Differentiate/integrate with explicit initial conditions; identify each force recipient and the inertial frame. Preserve force dimensions, geometric constraints and the distinction between model and solver error.
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For k=2 and requested x=3, the endpoint is exactly the first turn with v = 0 at x = 3 m. For k=6 the first turn is sqrt3 m, so the same target is inaccessible.
Open the experimentPredict
For k=4 and target x=2, the endpoint speed is:
Experiment
Derive the spatial relation from the time equation, test an exact turn and an inaccessible target, and explain where division by velocity would fail.
Use a spatial derivative only on an accessible branch
Worked teaching model · synthetic values · Virtual2 kg point starts x0=0 with v0=3 m/s under F=−k x. Integrate v dv/dx=−kx/m along the first rightward branch only. Report the turning endpoint if a requested target is inaccessible; do not describe later oscillations here.
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Squared speed on accessible first branch
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: 9−k x²/2 (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Squared speed on accessible first branch
9−k x²/2
| position (m) | speed squared (m²/s²) |
|---|---|
| 0 | 9 |
| 0.0265165 | 8.998594 |
| 0.05303301 | 8.994375 |
| 0.07954951 | 8.987344 |
| 0.106066 | 8.9775 |
| 0.1325825 | 8.964844 |
| 0.159099 | 8.949375 |
| 0.1856155 | 8.931094 |
| 0.212132 | 8.91 |
| 0.2386485 | 8.886094 |
| 0.265165 | 8.859375 |
| 0.2916815 | 8.829844 |
| 0.3181981 | 8.7975 |
| 0.3447146 | 8.762344 |
| 0.3712311 | 8.724375 |
| 0.3977476 | 8.683594 |
| 0.4242641 | 8.64 |
| 0.4507806 | 8.593594 |
| 0.4772971 | 8.544375 |
| 0.5038136 | 8.492344 |
| 0.5303301 | 8.4375 |
| 0.5568466 | 8.379844 |
| 0.5833631 | 8.319375 |
| 0.6098796 | 8.256094 |
| 0.6363961 | 8.19 |
| 0.6629126 | 8.121094 |
| 0.6894291 | 8.049375 |
| 0.7159456 | 7.974844 |
| 0.7424621 | 7.8975 |
| 0.7689786 | 7.817344 |
| 0.7954951 | 7.734375 |
| 0.8220116 | 7.648594 |
| 0.8485281 | 7.56 |
| 0.8750446 | 7.468594 |
| 0.9015611 | 7.374375 |
| 0.9280777 | 7.277344 |
| 0.9545942 | 7.1775 |
| 0.9811107 | 7.074844 |
| 1.007627 | 6.969375 |
| 1.034144 | 6.861094 |
| 1.06066 | 6.75 |
| 1.087177 | 6.636094 |
| 1.113693 | 6.519375 |
| 1.14021 | 6.399844 |
| 1.166726 | 6.2775 |
| 1.193243 | 6.152344 |
| 1.219759 | 6.024375 |
| 1.246276 | 5.893594 |
| 1.272792 | 5.76 |
| 1.299309 | 5.623594 |
| 1.325825 | 5.484375 |
| 1.352342 | 5.342344 |
| 1.378858 | 5.1975 |
| 1.405375 | 5.049844 |
| 1.431891 | 4.899375 |
| 1.458408 | 4.746094 |
| 1.484924 | 4.59 |
| 1.511441 | 4.431094 |
| 1.537957 | 4.269375 |
| 1.564474 | 4.104844 |
| 1.59099 | 3.9375 |
| 1.617507 | 3.767344 |
| 1.644023 | 3.594375 |
| 1.67054 | 3.418594 |
| 1.697056 | 3.24 |
| 1.723573 | 3.058594 |
| 1.750089 | 2.874375 |
| 1.776606 | 2.687344 |
| 1.803122 | 2.4975 |
| 1.829639 | 2.304844 |
| 1.856155 | 2.109375 |
| 1.882672 | 1.911094 |
| 1.909188 | 1.71 |
| 1.935705 | 1.506094 |
| 1.962221 | 1.299375 |
| 1.988738 | 1.089844 |
| 2.015254 | 0.8775 |
| 2.041771 | 0.6623437 |
| 2.068287 | 0.444375 |
| 2.094804 | 0.2235938 |
| 2.12132 | 1.77636e-15 |
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 |
|---|---|---|
| First rightward turning position | 2.12132 | m |
| Requested target reachable | 1 | flag |
| Actual endpoint position | 2 | m |
| Endpoint speed on rightward branch | 1 | m/s |
| Endpoint signed force | -8 | N |
The spatial form of acceleration is useful only with a checked branch and endpoint.
On a monotonic segment, dv/dt=(dv/dx)(dx/dt)=v dv/dx. For m2 and F=−kx, integrate v dv=−(k/2)x dx to get v²=9−k x²/2. The first zero is sqrt(18/k); farther targets are inaccessible on this branch. Dividing by v at the turning point would be singular, although the original time-domain force equation remains regular.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The spatial form of acceleration is useful only with a checked branch and endpoint.
On a monotonic segment, dv/dt=(dv/dx)(dx/dt)=v dv/dx. For m2 and F=−kx, integrate v dv=−(k/2)x dx to get v²=9−k x²/2. The first zero is sqrt(18/k); farther targets are inaccessible on this branch. Dividing by v at the turning point would be singular, although the original time-domain force equation remains regular.
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Why can the original differential equation remain regular at a spatial-coordinate turning singularity?
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