Keep the zero-energy escape limit distinct from finite turning
For radial outward motion in the supplied μ = 1 field, angular momentum is zero and specific energy is v²/2 − 1/r. The potential reference is zero at infinity. Escape speed at starting radius r₀ is sqrt(2/r₀), the speed that makes this energy exactly zero.
Learning outcome: Derive and compare sub-escape, exact-escape and above-escape branches, retaining the infinity reference and the absent surface model.
Starting recap
Integrate mass moments about named axes, derive torque and angular-momentum constraints, and test orbital energy, angular and numerical consistency against independent evidence.
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At r₀ = 2 m and f = 1.5, energy is 0.625 m²/s² and asymptotic speed is sqrt(1.25) ≈ 1.11803 m/s. This differs from the initial speed 1.5 m/s.
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At exactly the local escape speed, the ideal radial motion has:
Experiment
Derive and compare sub-escape, exact-escape and above-escape branches, retaining the infinity reference and the absent surface model.
Distinguish finite turning radius from escape energy
Worked teaching model · synthetic values · Radial outward test-particle motion with zero angular momentum in mu=1 field. Speed is a selected fraction of sqrt(2mu/r0), so fraction1 is an exact analytic zero-energy boundary. No physical launch, central surface or inward collision model.
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Inverse-square potential reference at infinity
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: −1/r (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Inverse-square potential reference at infinity
−1/r
| radius (m) | specific potential (m²/s²) |
|---|---|
| 2 | -0.5 |
| 2.125 | -0.4705882 |
| 2.25 | -0.4444444 |
| 2.375 | -0.4210526 |
| 2.5 | -0.4 |
| 2.625 | -0.3809524 |
| 2.75 | -0.3636364 |
| 2.875 | -0.3478261 |
| 3 | -0.3333333 |
| 3.125 | -0.32 |
| 3.25 | -0.3076923 |
| 3.375 | -0.2962963 |
| 3.5 | -0.2857143 |
| 3.625 | -0.2758621 |
| 3.75 | -0.2666667 |
| 3.875 | -0.2580645 |
| 4 | -0.25 |
| 4.125 | -0.2424242 |
| 4.25 | -0.2352941 |
| 4.375 | -0.2285714 |
| 4.5 | -0.2222222 |
| 4.625 | -0.2162162 |
| 4.75 | -0.2105263 |
| 4.875 | -0.2051282 |
| 5 | -0.2 |
| 5.125 | -0.195122 |
| 5.25 | -0.1904762 |
| 5.375 | -0.1860465 |
| 5.5 | -0.1818182 |
| 5.625 | -0.1777778 |
| 5.75 | -0.173913 |
| 5.875 | -0.1702128 |
| 6 | -0.1666667 |
| 6.125 | -0.1632653 |
| 6.25 | -0.16 |
| 6.375 | -0.1568627 |
| 6.5 | -0.1538462 |
| 6.625 | -0.1509434 |
| 6.75 | -0.1481481 |
| 6.875 | -0.1454545 |
| 7 | -0.1428571 |
| 7.125 | -0.1403509 |
| 7.25 | -0.137931 |
| 7.375 | -0.1355932 |
| 7.5 | -0.1333333 |
| 7.625 | -0.1311475 |
| 7.75 | -0.1290323 |
| 7.875 | -0.1269841 |
| 8 | -0.125 |
| 8.125 | -0.1230769 |
| 8.25 | -0.1212121 |
| 8.375 | -0.119403 |
| 8.5 | -0.1176471 |
| 8.625 | -0.115942 |
| 8.75 | -0.1142857 |
| 8.875 | -0.1126761 |
| 9 | -0.1111111 |
| 9.125 | -0.109589 |
| 9.25 | -0.1081081 |
| 9.375 | -0.1066667 |
| 9.5 | -0.1052632 |
| 9.625 | -0.1038961 |
| 9.75 | -0.1025641 |
| 9.875 | -0.1012658 |
| 10 | -0.1 |
| 10.125 | -0.09876543 |
| 10.25 | -0.09756098 |
| 10.375 | -0.09638554 |
| 10.5 | -0.0952381 |
| 10.625 | -0.09411765 |
| 10.75 | -0.09302326 |
| 10.875 | -0.09195402 |
| 11 | -0.09090909 |
| 11.125 | -0.08988764 |
| 11.25 | -0.08888889 |
| 11.375 | -0.08791209 |
| 11.5 | -0.08695652 |
| 11.625 | -0.08602151 |
| 11.75 | -0.08510638 |
| 11.875 | -0.08421053 |
| 12 | -0.08333333 |
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 |
|---|---|---|
| Local escape speed | 1 | m/s |
| Initial outward speed | 1 | m/s |
| Specific total energy | 0 | m²/s² |
| Unbound or marginally unbound | 1 | flag |
| Asymptotic outward speed | 0 | m/s |
Nonnegative energy has no finite outer turning radius; the zero-energy limit approaches zero speed only at infinity.
With U(infinity)=0, E=v0²/2−mu/r0. The speed fraction control makes the marginal boundary exact, E=0 at fraction1. Below escape, set final speed0 to find rmax=r0/(1−fraction²). Reaching arbitrarily large radius does not occur in finite time. No real launch or surface-impact prediction is made.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Nonnegative energy has no finite outer turning radius; the zero-energy limit approaches zero speed only at infinity.
With U(infinity)=0, E=v0²/2−mu/r0. The speed fraction control makes the marginal boundary exact, E=0 at fraction1. Below escape, set final speed0 to find rmax=r0/(1−fraction²). Reaching arbitrarily large radius does not occur in finite time. No real launch or surface-impact prediction is made.
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