Carry position and velocity across a contact-loss event
A support begins at rest with upward coordinate ys=−k t³/6, so its acceleration is−kt. A2 kg point initially rests on it. If contact holds, the point must share this acceleration. Newton's law gives N−mg=−mkt, or N=m(g−kt). A contact surface can push upward but cannot pull the point down.
Learning outcome: Locate separation analytically, compare a pre-event, exact-boundary and post-event state, and retain continuous position/velocity plus the growing support gap.
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=10, separation occurs at1 s. At t=2, point height is−35/3≈−11.6667 m while support height is−40/3≈−13.3333 m; gap5/3 m.
Open the experimentPredict
Once the required normal becomes negative, the model must:
Experiment
Locate separation analytically, compare a pre-event, exact-boundary and post-event state, and retain continuous position/velocity plus the growing support gap.
Switch equations at a unilateral-contact event
Worked teaching model · synthetic values · Virtual2 kg point initially rests on a support with ysupport=−k t³/6, upward positive. Contact can push upward but not pull. Before separation it shares support motion; afterward only gravity−10 m/s² acts. No collisions or recontact occur on this bounded downward-accelerating branch.
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Contact trajectory and rejected continuation
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: actual point (solid).2: support (dashed). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Contact trajectory and rejected continuation
actual point
| time (s) | height (m) |
|---|---|
| 0 | 0 |
| 0.025 | -1.56250e-5 |
| 0.05 | -0.000125 |
| 0.075 | -0.000421875 |
| 0.1 | -0.001 |
| 0.125 | -0.001953125 |
| 0.15 | -0.003375 |
| 0.175 | -0.005359375 |
| 0.2 | -0.008 |
| 0.225 | -0.01139063 |
| 0.25 | -0.015625 |
| 0.275 | -0.02079688 |
| 0.3 | -0.027 |
| 0.325 | -0.03432813 |
| 0.35 | -0.042875 |
| 0.375 | -0.05273438 |
| 0.4 | -0.064 |
| 0.425 | -0.07676562 |
| 0.45 | -0.091125 |
| 0.475 | -0.1071719 |
| 0.5 | -0.125 |
| 0.525 | -0.1447031 |
| 0.55 | -0.166375 |
| 0.575 | -0.1901094 |
| 0.6 | -0.216 |
| 0.625 | -0.2441406 |
| 0.65 | -0.274625 |
| 0.675 | -0.3075469 |
| 0.7 | -0.343 |
| 0.725 | -0.3810781 |
| 0.75 | -0.421875 |
| 0.775 | -0.4654844 |
| 0.8 | -0.512 |
| 0.825 | -0.5615156 |
| 0.85 | -0.614125 |
| 0.875 | -0.6699219 |
| 0.9 | -0.729 |
| 0.925 | -0.7914531 |
| 0.95 | -0.857375 |
| 0.975 | -0.9268594 |
| 1 | -1 |
| 1.025 | -1.076891 |
| 1.05 | -1.157625 |
| 1.075 | -1.242297 |
| 1.1 | -1.331 |
| 1.125 | -1.423828 |
| 1.15 | -1.520875 |
| 1.175 | -1.622234 |
| 1.2 | -1.728 |
| 1.225 | -1.838266 |
| 1.25 | -1.953125 |
| 1.275 | -2.072672 |
| 1.3 | -2.197 |
| 1.325 | -2.326203 |
| 1.35 | -2.460375 |
| 1.375 | -2.599609 |
| 1.4 | -2.744 |
| 1.425 | -2.893641 |
| 1.45 | -3.048625 |
| 1.475 | -3.209047 |
| 1.5 | -3.375 |
| 1.525 | -3.546578 |
| 1.55 | -3.723875 |
| 1.575 | -3.906984 |
| 1.6 | -4.096 |
| 1.625 | -4.291016 |
| 1.65 | -4.492125 |
| 1.675 | -4.699421 |
| 1.7 | -4.912963 |
| 1.725 | -5.132755 |
| 1.75 | -5.358796 |
| 1.775 | -5.591088 |
| 1.8 | -5.82963 |
| 1.825 | -6.074421 |
| 1.85 | -6.325463 |
| 1.875 | -6.582755 |
| 1.9 | -6.846296 |
| 1.925 | -7.116088 |
| 1.95 | -7.39213 |
| 1.975 | -7.674421 |
| 2 | -7.962963 |
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.
support
| time (s) | height (m) |
|---|---|
| 0 | 0 |
| 0.025 | -1.56250e-5 |
| 0.05 | -0.000125 |
| 0.075 | -0.000421875 |
| 0.1 | -0.001 |
| 0.125 | -0.001953125 |
| 0.15 | -0.003375 |
| 0.175 | -0.005359375 |
| 0.2 | -0.008 |
| 0.225 | -0.01139063 |
| 0.25 | -0.015625 |
| 0.275 | -0.02079688 |
| 0.3 | -0.027 |
| 0.325 | -0.03432813 |
| 0.35 | -0.042875 |
| 0.375 | -0.05273438 |
| 0.4 | -0.064 |
| 0.425 | -0.07676562 |
| 0.45 | -0.091125 |
| 0.475 | -0.1071719 |
| 0.5 | -0.125 |
| 0.525 | -0.1447031 |
| 0.55 | -0.166375 |
| 0.575 | -0.1901094 |
| 0.6 | -0.216 |
| 0.625 | -0.2441406 |
| 0.65 | -0.274625 |
| 0.675 | -0.3075469 |
| 0.7 | -0.343 |
| 0.725 | -0.3810781 |
| 0.75 | -0.421875 |
| 0.775 | -0.4654844 |
| 0.8 | -0.512 |
| 0.825 | -0.5615156 |
| 0.85 | -0.614125 |
| 0.875 | -0.6699219 |
| 0.9 | -0.729 |
| 0.925 | -0.7914531 |
| 0.95 | -0.857375 |
| 0.975 | -0.9268594 |
| 1 | -1 |
| 1.025 | -1.076891 |
| 1.05 | -1.157625 |
| 1.075 | -1.242297 |
| 1.1 | -1.331 |
| 1.125 | -1.423828 |
| 1.15 | -1.520875 |
| 1.175 | -1.622234 |
| 1.2 | -1.728 |
| 1.225 | -1.838266 |
| 1.25 | -1.953125 |
| 1.275 | -2.072672 |
| 1.3 | -2.197 |
| 1.325 | -2.326203 |
| 1.35 | -2.460375 |
| 1.375 | -2.599609 |
| 1.4 | -2.744 |
| 1.425 | -2.893641 |
| 1.45 | -3.048625 |
| 1.475 | -3.209047 |
| 1.5 | -3.375 |
| 1.525 | -3.546578 |
| 1.55 | -3.723875 |
| 1.575 | -3.906984 |
| 1.6 | -4.096 |
| 1.625 | -4.291016 |
| 1.65 | -4.492125 |
| 1.675 | -4.699422 |
| 1.7 | -4.913 |
| 1.725 | -5.132953 |
| 1.75 | -5.359375 |
| 1.775 | -5.592359 |
| 1.8 | -5.832 |
| 1.825 | -6.078391 |
| 1.85 | -6.331625 |
| 1.875 | -6.591797 |
| 1.9 | -6.859 |
| 1.925 | -7.133328 |
| 1.95 | -7.414875 |
| 1.975 | -7.703734 |
| 2 | -8 |
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 separation time | 1.666667 | s |
| Actual point height | -7.962963 | m |
| Support height | -8 | m |
| Point above support gap | 0.03703704 | m |
| Actual vertical velocity | -11.66667 | m/s |
| Actual upward normal force | 0 | N |
| Actual vertical acceleration | -10 | m/s² |
The required pulling normal is impossible; the point follows a freefall branch after separation.
While contact holds, N−mg=m(−k t), so N=m(g−k t). Its first zero is g/k. Carry continuous position and velocity from that event into freefall rather than resetting either to zero. At the exact boundary acceleration agrees on both sides; afterward the support falls away faster and the gap grows. A negative algebraic normal is a failed contact assumption, not a physical pulling surface.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The required pulling normal is impossible; the point follows a freefall branch after separation.
While contact holds, N−mg=m(−k t), so N=m(g−k t). Its first zero is g/k. Carry continuous position and velocity from that event into freefall rather than resetting either to zero. At the exact boundary acceleration agrees on both sides; afterward the support falls away faster and the gap grows. A negative algebraic normal is a failed contact assumption, not a physical pulling surface.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
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