SCIENCE · INVESTIGATE AND EXPLAIN

Subtract common acceleration before interpreting a tidal effect

Gravity accelerates the whole body toward a source, but it also varies across the body. A tidal effect concerns that spatial difference rather than the common acceleration. In a freely falling center frame, the near side is pulled toward the source more strongly than the center, while the far side falls behind it. Both inertial accelerations still point toward the source; the opposite relative signs do not require a repulsive gravitational force.

Learning outcome: Compare exact endpoint differences with the small-body approximation at different D and R. Retain common and relative accelerations separately, and identify the missing material information needed to predict deformation.

Starting recap

Retain relative versus barycentric geometry, time versus angle, fitted versus instantaneous velocity and complete energy/angular-momentum boundaries.

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Try

At D=10 and R=0.2, near-minus-center is≈0.000412328 and far-minus-center≈−0.000388312. Full stretching is≈0.000800640, close to the leading 4R/D³=0.0008. The slight asymmetry reflects finite R/D, not opposite gravitational force directions.

Open the experiment

Predict

Why can the near and far center-relative accelerations have opposite signs when both are attracted toward the source?

Your prediction and reason

Experiment

Compare exact endpoint differences with the small-body approximation at different D and R. Retain common and relative accelerations separately, and identify the missing material information needed to predict deformation.

Subtract center acceleration before interpreting tidal stretching

Worked teaching model · synthetic values · Original source μ=1 and three collinear test points at D−R, D and D+R. Accelerations toward source are inverse-square; near-minus-far stretch compared with leading small-body approximation 4R/D³. No deformation, ocean dynamics, self-gravity or tide-height prediction.

Center separation from source: 10 normalized length Radial half-length of test body: 0.2 normalized length

Reset keeps your prediction, captures and practice history.

Common free fall and differential stretching are different gravitational effects.

All three inertial accelerations point toward the source, yet center-relative accelerations have opposite signs. The leading differential term scales as distance cubed in the denominator, unlike the inverse-square center acceleration. Actual tides require material response and dynamics.

Original conditional quantities and explicit units
Quantity	Value	Unit
Near-point acceleration toward source	0.01041233	length/time²
Center acceleration toward source	0.01000000	length/time²
Far-point acceleration toward source	0.009611688	length/time²
Near relative to center	0.0004123282	length/time²
Far relative to center	-0.0003883122	length/time²
Exact near-minus-far stretching	0.0008006404	length/time²
Leading small-body approximation	0.0008000000	length/time²
Approximation relative error	-0.0007998400	fraction

These original models are not measured orbit records or operational mission instructions. Retain the frame, assumptions and missing evidence before interpreting a real system.

Common free fall and differential stretching are different gravitational effects.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Define the orbital evidence

2. Connect geometry, force and conservation

The original collinear test points have source parameter μ=1, center distance D and radial half-length R. Their source-directed accelerations are 1/(D−R)², 1/D² and 1/(D+R)². Subtract the center value from each side to obtain relative acceleration. The full near-minus-far stretching is the difference of the endpoint values, not the center acceleration itself.

For R much smaller than D, expanding the inverse-square field gives a full endpoint difference approximately 4R/D³. Each side relative to the center has leading magnitude 2R/D³. This distinction prevents a factor-of-two error, and the inverse-cube distance dependence differs from the inverse-square common acceleration. The exact finite difference remains available to test the approximation.

The test points do not constitute a physical ocean or elastic planet. Real deformation depends on self-gravity, rigidity, rotation and time-dependent response; local tides also depend on basin geometry and dynamics. No tide height or hazard follows directly from this acceleration ledger. The source mechanism provides context, while all numerical distances and responses here are original normalized examples.

3. Work the original case

4. Predict and investigate

Why can the near and far center-relative accelerations have opposite signs when both are attracted toward the source?

Apply the evidence to a changed case

Keep R=0.2 and reduce D from 10 to 5. Compare leading full stretching and center acceleration factors.

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

Leading stretching increases from 0.0008 to 0.0064, a factor of 8, while center acceleration rises from 0.01 to 0.04, a factor of 4. The exact D=5 endpoint difference≈0.00642053 is slightly above its leading approximation.

Source and boundaries

Primary mechanism reference. Checked 30 October 2026. Original prose, numerical cases and diagrams; source context is not a claim that these fictional records were observed. No actual mission operation or instrument aiming is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Keep R=0.2 and reduce D from 10 to 5. Compare leading full stretching and center acceleration factors.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

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Previous: Distinguish reaching a radius from joining its circular orbit Next: Close the torque and work boundary of an orbital change

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