Preview

Gravitational Fields

Q: At 3 AU from a 10 M⊕ body the panel reads g = 1.979×10⁻⁸ m/s². What will it read at 1.5 AU — and why is it not simply doubled?

G = GM/r² — halving r multiplies g by 4: the panel reads 7.915×10⁻⁸ m/s². Check V too: it only doubles (−GM/r pays the distance once).

Q: The panel shows V = −8.881×10⁻³ MJ/kg and Binding E = −8.881×10³ J at the default state. Why are the two numbers the same?

Binding energy is computed for a 1 kg test mass: U = −GMm/r with m = 1 equals V numerically — the panel is a per-kilogram ledger in two currencies.

Q: Escape velocity from Earth is ~11.2 km/s. What is the ratio v_esc / v_circ?

V_esc = √(2GM/R) = √2 × v_circ.

Q: The Near Black Hole preset reads an escape speed of just 1.03 km/s. How small would 100 M⊕ have to be packed for its escape speed to actually reach c?

You can work it out this way: set v_esc = c in √(2GM/r): r = 2GM/c² ≈ 0.89 m. A real black hole is defined by its escape speed, not its costume.

Field strength, potential wells, and escape velocity

In 1666 Newton asked whether the pull that drops an apple could reach the Moon. The Moon sits 60 Earth-radii away, and its centripetal acceleration came out almost exactly 1/3600 of surface g — 60 squared. One law for falling apples and falling moons, and the inverse-square signature was born. This lab hands you that law as a landscape. A central mass prices every point of space around it: the field strength g = GM/r² tells you how hard space pulls at that address, and the potential V = −GM/r tells you how deep the well is there. Both are live on the panel, and they do not fade at the same rate — double your distance and the pull quarters while the depth only halves. That difference is the whole game. The well picture pays off at the escape threshold: climb out until your kinetic energy exactly cancels the negative potential, and your total energy is zero — that speed, √(2GM/r), is the escape velocity, and it does not care how massive your ship is. You will also meet two honest surprises. The preset named 'Earth Surface' parks you a full AU away, where g is about two billionths of 9.81 — names are not physics. And the 'black hole' preset, for all its purple glow, prices an escape speed of barely 1 km/s — a real black hole needs the escape speed to reach light itself. The field never switches off, the well never bottoms out at zero, and every number on the panel is yours to predict before you touch a slider.

What you'll be able to do

Price the inverse-square field on the panel's own numbers: at the default state (10 M_E, 3 AU) g = GM/r² = 1.978793e-8 → row '1.979e-8 m/s²'; doubling the distance (3 → 6 AU) QUARTERS it to 4.946982e-9 ('4.947e-9'), and the full-domain sweep collapses g by exactly 400× (7.123654e-7 at 0.5 AU → 1.780913e-9 at 10 AU, (10/0.5)²) — Newton's Moon test as a slider: 'the Moon, at a distance of about 60R_E, has a centripetal acceleration about (60)² times smaller than g' (OpenStax UP1 §13.1, verified), because 'the surface area of that sphere is proportional to r²'
Read the potential well as a signed depth, not a strength: V = −GM/r is always negative and climbs TOWARD zero as r grows ('U→0 as r→∞', 'U becomes increasingly more negative as the masses get closer' — OpenStax UP1 §13.3, verified) — at defaults −8880.8217 J/kg ('−8.881e-3 MJ/kg'), doubling r HALVES the depth to −4440.4109 ('−4.440e-3') while g quarters: same slider, two exponents. The Binding E row is the same book per kilogram (mTest = 1 kg hardcoded: −8.881e3 J ≡ −8.881e-3 MJ/kg, registered redundancy), and 'deeper in the well = more tightly bound' is the panel's own arithmetic
Own the escape threshold as an energy statement, not a distance: v_esc = √(2GM/r) is 'the speed at any position such that the total energy is zero' (OpenStax UP1 §13.3, verified), with 'm canceled out… the same for all objects, regardless of mass' (verbatim — misconception 1's counter); on the panel the √M and 1/√r scalings are live (Central Mass 1 → 100 at 3 AU: v_esc 42.144565 → 421.44565 m/s, ×10 for ×100; distance ×4 → v_esc ÷2, the HTML quiz Q3's law). The 'Near Black Hole' preset prices v_esc = 1.03 km/s at (100 M_E, 0.5 AU) — 0.0003% of c, since 100 Earth masses would be a black hole only if squeezed inside its 0.886 m Schwarzschild radius (node-verified): the purple glow is a costume, the field law underneath is still Newton's

Formulas

F = G\frac{m_1 m_2}{r^2}

Newton's law of universal gravitation

g = \frac{GM}{r^2}

Gravitational field strength at distance r from mass M

V = -\frac{GM}{r}

Gravitational potential (energy per unit mass)

v_{esc} = \sqrt{\frac{2GM}{r}}

Escape velocity — minimum speed to leave the field

U = -\frac{GMm}{r}

Gravitational PE for a test mass m at distance r

Make a prediction

Default state: Central Mass 10 M_E, Test Distance 3 AU — the panel reads g = 1.979e-8 m/s² and V = −8.881e-3 MJ/kg. You drag Test Distance out to 6 AU, exactly twice as far from the central mass. What do the g and V rows read now?

No grading here — commit to a guess, then scroll down and test it yourself.

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Run the prediction live from the defaults: drag Test Distance 3 → 6 AU and read the two rows — g flips 1.979e-8 → 4.947e-9 m/s² (÷4) while V flips −8.881e-3 → −4.440e-3 MJ/kg (÷2, less negative). Same drag, two rates. Then drag back to 1.5 AU (half the distance): g quadruples to '7.915e-8' and V doubles in depth to '−1.776e-2'. The square and the single power, both settled by your own slider.
Click the three preset buttons in turn with the panel as your guide: Earth — catch the misnomer (named 'Earth Surface', but the g row reads 1.781e-8 m/s² at a full AU, not 9.81); Jupiter — catch the clamp (the mass label reads 100.0 M_E, not the 318 the preset asked for) and read g = 6.586e-8; Near Black Hole — read v_esc = 1.03 km/s and ask what a real black hole would need. Names and costumes are not physics; the rows are.
Audit the escape threshold by hand: at the Near Black Hole preset, read v_esc = 1.03 km/s and compare it with Earth's real 11.2 km/s — this 'black hole' is 0.0003% of light speed from earning its name. Then hold Test Distance at 3 AU and sweep Central Mass 1 → 100: v_esc climbs 42.14 → 421.45 m/s, exactly ×10 for ×100 mass — the √M signature. Notice what is missing: no ship-mass control exists anywhere, because 'm has canceled out of the equation.'
Take the inverse-square across its whole domain: hold Central Mass at 10 and sweep Test Distance 0.5 → 10 AU, reading the g row as you go — it collapses from 7.124e-7 to 1.781e-9 m/s², a 400× fall for a 20× stretch, while the V row only falls 20× (−5.328e-2 → −2.664e-3 MJ/kg). Watch the arrow field shrink and the well shallow with it. Then try to find zero: bottom the mass at 1 M_E, push distance to 10 AU — g still reads 1.781e-10. The field ends only at infinity.

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Check

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Your prediction

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Answer: g = 4.947e-9 m/s² and V = −4.440e-3 MJ/kg — g pays the distance SQUARED (÷4), the potential pays it only once (÷2, climbing toward zero from below)

The two rows obey two different laws, and one drag exposes both. The field strength is g = GM/r²: doubling r from 3 to 6 AU divides g by 4, from 1.978793e-8 to 4.946982e-9 — the panel shows '4.947e-9 m/s²'. The square is geometry, not a fudge: 'The strength of any source at a distance r is spread over the surface of a sphere centered about the mass. The surface area of that sphere is proportional to r²' (OpenStax UP1 §13.1). The potential is V = −GM/r: doubling r divides only once, from −8880.8217 to −4440.4109 J/kg — the panel shows '−4.440e-3 MJ/kg', less negative, climbing toward the zero it reaches only at infinity ('U→0 as r→∞', §13.3). Option B is the linear instinct — it prices V correctly by accident and g wrongly, halving what should quarter; it is the exact trap the sim's own quiz Q1 sets with its '1/2 as strong' distractor, and Newton's Moon test is the historical refutation (gravity at 60 Earth-radii is 1/3600 of surface g, not 1/60). Option C is the zero-gravity-in-space misconception outright — and the field itself disagrees: gravity reaches the Moon and still steers it (§13.1), g at 400 km altitude is 8.67 m/s², 88% of surface (§13.3), and in this sim the g row never reads zero anywhere in the domain — even 1 M_E at 10 AU reads 1.781e-10 m/s². Astronauts float because they fall, not because gravity is gone. Watch the two rows as you drag: they move at different rates, and neither ever switches off.

Quiz (0/4)

At 3 AU from a 10 M⊕ body the panel reads g = 1.979×10⁻⁸ m/s². What will it read at 1.5 AU — and why is it not simply doubled?

The panel shows V = −8.881×10⁻³ MJ/kg and Binding E = −8.881×10³ J at the default state. Why are the two numbers the same?

Escape velocity from Earth is ~11.2 km/s. What is the ratio v_esc / v_circ?

The Near Black Hole preset reads an escape speed of just 1.03 km/s. How small would 100 M⊕ have to be packed for its escape speed to actually reach c?

You can now

Price the inverse-square field on the panel's own numbers: at the default state (10 M_E, 3 AU) g = GM/r² = 1.978793e-8 → row '1.979e-8 m/s²'; doubling the distance (3 → 6 AU) QUARTERS it to 4.946982e-9 ('4.947e-9'), and the full-domain sweep collapses g by exactly 400× (7.123654e-7 at 0.5 AU → 1.780913e-9 at 10 AU, (10/0.5)²) — Newton's Moon test as a slider: 'the Moon, at a distance of about 60R_E, has a centripetal acceleration about (60)² times smaller than g' (OpenStax UP1 §13.1, verified), because 'the surface area of that sphere is proportional to r²'
Read the potential well as a signed depth, not a strength: V = −GM/r is always negative and climbs TOWARD zero as r grows ('U→0 as r→∞', 'U becomes increasingly more negative as the masses get closer' — OpenStax UP1 §13.3, verified) — at defaults −8880.8217 J/kg ('−8.881e-3 MJ/kg'), doubling r HALVES the depth to −4440.4109 ('−4.440e-3') while g quarters: same slider, two exponents. The Binding E row is the same book per kilogram (mTest = 1 kg hardcoded: −8.881e3 J ≡ −8.881e-3 MJ/kg, registered redundancy), and 'deeper in the well = more tightly bound' is the panel's own arithmetic
Own the escape threshold as an energy statement, not a distance: v_esc = √(2GM/r) is 'the speed at any position such that the total energy is zero' (OpenStax UP1 §13.3, verified), with 'm canceled out… the same for all objects, regardless of mass' (verbatim — misconception 1's counter); on the panel the √M and 1/√r scalings are live (Central Mass 1 → 100 at 3 AU: v_esc 42.144565 → 421.44565 m/s, ×10 for ×100; distance ×4 → v_esc ÷2, the HTML quiz Q3's law). The 'Near Black Hole' preset prices v_esc = 1.03 km/s at (100 M_E, 0.5 AU) — 0.0003% of c, since 100 Earth masses would be a black hole only if squeezed inside its 0.886 m Schwarzschild radius (node-verified): the purple glow is a costume, the field law underneath is still Newton's