Gravitational Force & Escape Velocity Calculator
Calculate the gravitational force between any two masses using Newton's Universal Law of Gravitation, F = Gm₁m₂/r². Or select a planet to instantly compute its escape velocity, surface gravity, and orbital speed.
Includes full formula derivations, 3 worked examples, a 10-body planetary comparison table, and definitions of every key term. Built by He Loves Math for physics students and curious minds.
The Two Core Formulas
Where \(G = 6.674 \times 10^{-11}\ \text{N·m}^2/\text{kg}^2\), \(m_1, m_2\) are masses in kg, \(r\) is the centre-to-centre separation in metres, \(M\) is the body's mass, and \(r\) is the body's radius for escape velocity.
Physics Calculator
Gravitational Force
—What Is Gravitational Force?
Gravitational force is the universal attractive force that acts between every two objects with mass in the universe. It is the force that keeps your feet on the ground, holds the Moon in orbit around Earth, guides the planets around the Sun, binds galaxies together across billions of light-years, and is responsible for the large-scale structure of the cosmos — galaxy clusters, filaments, and cosmic voids.
Gravity is one of four fundamental forces of nature: (1) strong nuclear force (holds atomic nuclei together), (2) weak nuclear force (responsible for radioactive decay), (3) electromagnetic force (holds atoms and molecules together), and (4) gravity. At the human scale, gravity feels powerful — it governs all of everyday mechanics. Yet quantitatively, gravity is by far the weakest of the four forces. The gravitational attraction between two protons is roughly 10³⁶ times weaker than the electromagnetic repulsion between them.
This weakness is why the gravitational force between two small objects — like two 1 kg spheres placed 1 metre apart — is only about 6.67 × 10⁻¹¹ N, a force so negligible it cannot be felt. Only when you accumulate the mass of entire planets does gravity become the dominant force governing dynamics.
Newton's Law of Universal Gravitation & the Constant G
Published in 1687 in Isaac Newton's Philosophiæ Naturalis Principia Mathematica — widely regarded as one of the greatest scientific works ever written — Newton's Law of Universal Gravitation gives a precise, quantitative description of gravitational force between any two point masses:
Where:
- \(F\) = gravitational force (Newtons, N)
- \(G\) = gravitational constant = \(6.674 \times 10^{-11}\ \text{N·m}^2/\text{kg}^2\)
- \(m_1, m_2\) = masses of the two objects (kilograms)
- \(r\) = distance between the centres of mass (metres)
The gravitational constant G is one of the fundamental constants of nature — it appears in every gravitational calculation in the universe. Newton himself did not have a numerical value for G; he understood the law's structure but could not measure G directly. The first measurement came over a century later: in 1798, Henry Cavendish used an exquisitely sensitive torsion balance to measure the tiny attractive force between lead spheres, deriving G to within about 1% of today's value. This experiment also gave us the first measurement of Earth's mean density and total mass.
The current internationally accepted value (CODATA 2018) is:
G is among the least precisely known fundamental constants — its relative uncertainty is about 2.2 × 10⁻⁵ — because gravity is so weak that measuring G requires isolating objects from all other disturbances, a formidable experimental challenge.
The Inverse-Square Law
The \(1/r^2\) dependence in Newton's law is called the inverse-square law. It arises because gravitational influence propagates outward in three-dimensional space: as distance doubles, that influence spreads over a sphere whose surface area is 4 times larger (area = 4πr²), so the force per unit area drops by a factor of 4.
Escape Velocity — Derivation from Energy Conservation
Escape velocity is the minimum launch speed needed so that an object can, in principle, travel infinitely far from a body's surface — reaching "infinity" with zero remaining kinetic energy. No further thrust is applied after launch.
The derivation uses conservation of total mechanical energy. At the surface (radius r), the object has kinetic energy KE = ½mv² and gravitational potential energy:
At infinity (\(r \to \infty\)), both KE and PE approach zero if the object just barely escapes. Setting total energy to zero:
The mass \(m\) of the escaping object cancels — escape velocity is independent of the mass of the escaping object. Solving for \(v_e\):
This can also be written in terms of surface gravitational acceleration \(g = GM/r^2\):
For Earth: \(g = 9.81\ \text{m/s}^2\), \(r = 6.371 \times 10^6\ \text{m}\), so \(v_e = \sqrt{2 \times 9.81 \times 6.371 \times 10^6} = \sqrt{1.250 \times 10^8} \approx 11{,}180\ \text{m/s} = 11.2\ \text{km/s}\).
3 Fully Worked Examples
Example 1 — Gravitational Force Between Earth and Moon
Given: \(m_1 = 5.972 \times 10^{24}\ \text{kg}\) (Earth), \(m_2 = 7.342 \times 10^{22}\ \text{kg}\) (Moon), \(r = 3.844 \times 10^8\ \text{m}\) (mean centre-to-centre distance)
Solution:
Interpretation: ~1.98 × 10²⁰ N — approximately 2 × 10¹⁹ tonnes-force. This immense force holds the Moon in its orbit and is the primary driver of Earth's ocean tides. Despite being so large, it is entirely negligible compared to the Earth–Sun force.
Example 2 — Force Between Two 1 kg Spheres 1 Metre Apart
Given: \(m_1 = m_2 = 1\ \text{kg}\), \(r = 1\ \text{m}\)
Solution:
Interpretation: 6.67 × 10⁻¹¹ N is the gravitational force between two 1 kg objects separated by 1 m — exactly numerically equal to G itself. This is 100 billion times smaller than 1 N (the weight of a 102 g object). It would require a precision torsion balance like Cavendish's to detect this force. This is why gravity between ordinary objects is completely unnoticeable in daily life.
Example 3 — Escape Velocity from Earth's Surface
Given: \(M_{\oplus} = 5.972 \times 10^{24}\ \text{kg}\), \(R_{\oplus} = 6.371 \times 10^6\ \text{m}\)
Solution:
Interpretation: 11.2 km/s ≈ 40,320 km/h ≈ 25,020 mph. This is about 33 times the speed of sound at sea level (343 m/s). Due to Earth's atmosphere, actual spacecraft must reach this speed above the atmosphere where drag is negligible. The Saturn V rocket that launched the Apollo missions achieved this in stages by burning fuel continuously (not in a single "cannon-ball" launch), which is why the actual escape trajectory can be slower as long as thrust continues.
Escape Velocity — Full Explanation
The term "escape velocity" was popularized in the 19th century, but the underlying physics was understood much earlier. In 1798, John Michell — the geologist who invented the torsion balance later used by Cavendish — calculated what he called "dark stars": bodies so massive and compact that even light could not escape. Jules Verne's 1865 novel From the Earth to the Moon described a cannon-launched spacecraft, which would require precisely the escape velocity at launch (no atmospheric consideration in the story).
There are three historically important "cosmic velocities":
- First cosmic velocity (orbital velocity): \(v_1 = \sqrt{GM/r}\) — the speed needed to maintain a circular orbit at the surface. For Earth: ~7.9 km/s. This is the minimum speed for a satellite in very low orbit.
- Second cosmic velocity (escape velocity): \(v_2 = \sqrt{2GM/r} = \sqrt{2}\ v_1\) — the speed needed to escape the body's gravitational field entirely. For Earth: ~11.2 km/s.
- Third cosmic velocity: the speed needed to escape the Solar System from Earth's orbit — approximately 16.7 km/s relative to Earth (combined effect of Earth's orbital velocity of 29.8 km/s).
Orbital Velocity vs Escape Velocity
The relationship between orbital velocity and escape velocity is elegant: escape velocity is always exactly \(\sqrt{2} \approx 1.414\) times the circular orbital velocity at the same radius. This follows directly from comparing their energy conditions: orbit requires ½mv² = ½ × (gravitational PE); escape requires ½mv² = gravitational PE.
Gravitational Potential Energy
The negative sign means the system is gravitationally bound. To separate two masses to infinity (U = 0) requires positive work to be done on the system. The escape energy per unit mass is exactly GM/r, and since kinetic energy = ½v², this gives v_e = √(2GM/r).
Surface Gravitational Acceleration
Surface gravity g (in m/s²) is simply the gravitational field strength at radius r. Earth's g = 9.81 m/s²; Moon's g = 1.62 m/s² (16.5% of Earth's). A 70 kg person weighs 686 N on Earth but only 113 N on the Moon — the same mass, 1/6 the weight.
Black Holes and the Schwarzschild Radius
A black hole forms when an object's escape velocity reaches or exceeds the speed of light, \(c = 2.998 \times 10^8\ \text{m/s}\). Setting \(v_e = c\) and solving for the critical radius gives the Schwarzschild radius, first derived by Karl Schwarzschild in 1916 from Einstein's general relativity equations:
Remarkably, this is exactly the result from Newtonian mechanics by setting \(v_e = c\) — though the full derivation requires general relativity to be rigorous. Key values:
- Earth (\(M = 5.972 \times 10^{24}\ \text{kg}\)): \(r_s \approx 8.87\ \text{mm}\) — compress Earth to a marble to form a black hole
- Sun (\(M = 1.989 \times 10^{30}\ \text{kg}\)): \(r_s \approx 2.95\ \text{km}\) — compress to a small town
- Sagittarius A* (4× 10⁶ M☉): \(r_s \approx 11.8 \times 10^6\ \text{km}\) ≈ 17 solar radii
The boundary at \(r_s\) is called the event horizon — the point of no return beyond which not even light can escape. Inside the event horizon, all paths in spacetime lead inevitably toward the singularity at the centre.
Planetary Data — Escape Velocities and Surface Gravity
| Body | Mass (kg) | Radius (km) | Surface g (m/s²) | Escape Velocity | Orbital Velocity (at surface) |
|---|---|---|---|---|---|
| Mercury | 3.301 × 10²³ | 2,440 | 3.70 | 4.25 km/s | 3.01 km/s |
| Venus | 4.867 × 10²⁴ | 6,051 | 8.87 | 10.36 km/s | 7.33 km/s |
| Earth | 5.972 × 10²⁴ | 6,371 | 9.81 | 11.19 km/s | 7.91 km/s |
| Moon | 7.342 × 10²² | 1,737 | 1.62 | 2.38 km/s | 1.68 km/s |
| Mars | 6.417 × 10²³ | 3,390 | 3.72 | 5.03 km/s | 3.56 km/s |
| Jupiter | 1.898 × 10²⁷ | 71,492 | 24.79 | 59.54 km/s | 42.11 km/s |
| Saturn | 5.683 × 10²⁶ | 60,268 | 10.44 | 35.49 km/s | 25.09 km/s |
| Uranus | 8.681 × 10²⁵ | 25,362 | 8.87 | 21.38 km/s | 15.12 km/s |
| Neptune | 1.024 × 10²⁶ | 24,622 | 11.15 | 23.56 km/s | 16.66 km/s |
| Sun | 1.989 × 10³⁰ | 695,700 | 274.0 | 617.5 km/s | 436.7 km/s |
| Neutron Star (~2M☉) | ~3.98 × 10³⁰ | ~10 | ~1.3 × 10¹² | ~1.9 × 10⁵ km/s (64% c) | ~1.3 × 10⁵ km/s |
Frequently Asked Questions
What is Newton's Law of Universal Gravitation?
Newton's Law states that every pair of masses attracts each other with a force \(F = Gm_1m_2/r^2\), where G is the gravitational constant (6.674 × 10⁻¹¹ N·m²/kg²), m₁ and m₂ are the masses, and r is the centre-to-centre distance. The law was published in 1687 and held as the definitive description of gravity for over 200 years. Einstein's 1915 General Theory of Relativity showed that Newton's law is an approximation — accurate for weak fields and slow velocities, but requiring correction near very massive or dense objects (like black holes, neutron stars, or Mercury's orbit close to the Sun).
What is the gravitational constant G and who measured it?
G = 6.674 × 10⁻¹¹ N·m²/kg² is a fundamental constant of nature connecting gravitational force to mass and distance. It was first measured by Henry Cavendish in 1798 using a torsion balance — a horizontal rod with lead spheres at each end suspended by a fine wire. When large lead masses were brought close, the tiny gravitational attraction twisted the wire by a measurable angle, allowing G (and from it, Earth's total mass and density) to be calculated. Cavendish's result was within 1% of today's accepted value. G is still the least precisely known fundamental constant — measuring it precisely is an active area of experimental physics.
What is escape velocity and how is it derived?
Escape velocity is the minimum speed to escape a celestial body's gravity without further thrust. It is derived from energy conservation: ½mv² = GMm/r → v_e = √(2GM/r). The mass of the escaping object cancels, so escape velocity is independent of it. Earth's escape velocity is approximately 11.2 km/s (40,320 km/h). It does not depend on launch direction — at exactly escape velocity, any direction (upward, sideways, even retrograde if there were no ground) would result in escape. In practice, rockets launch eastward to gain the benefit of Earth's rotation speed (up to ~460 m/s at the equator).
What is Earth's escape velocity?
Earth's escape velocity is ~11,186 m/s (11.19 km/s ≈ 40,280 km/h ≈ 25,020 mph). This was first calculated after Newton's law was established and G was measured. Interestingly, the Moon's escape velocity is only 2.38 km/s — about 1/4.7 of Earth's. This is why Apollo Lunar Module ascent stages could launch using a small engine (the Ascent Propulsion System, ~15 kN) whereas Saturn V had 34 million Newtons of thrust to escape Earth. The Moon's lower gravity (1.62 m/s²) and smaller radius combine to give a much gentler escape requirement.
Why is gravity the weakest force?
Gravity is approximately 10³⁶ times weaker than electromagnetism and vastly weaker than the strong and weak nuclear forces. The gravitational force between two protons is about 10⁻³⁶ of their electromagnetic repulsion. Yet gravity dominates at cosmic scales because: (1) it is always attractive (never repulsive like charges of the same sign), so it accumulates without limit; (2) it has infinite range; (3) it couples to all mass, including dark matter. On small scales, electromagnetic and nuclear forces completely overwhelm gravity — which is why gravity is irrelevant inside atoms. But on planetary, stellar, and galactic scales, gravity is the organizing force of the universe.
What is the Schwarzschild radius?
The Schwarzschild radius is the critical radius at which escape velocity equals the speed of light: r_s = 2GM/c². An object with all its mass inside r_s becomes a black hole — nothing, not even light, can escape. For the Sun: r_s ≈ 2.95 km (the Sun would need to shrink from its actual 695,700 km radius to less than 3 km to become a black hole). For Earth: r_s ≈ 8.87 mm. First derived by Karl Schwarzschild in 1916 from Einstein's General Theory of Relativity, the Schwarzschild radius defines the event horizon of a non-rotating black hole.
How does the inverse-square law work?
The inverse-square law means F ∝ 1/r²: doubling the distance reduces gravitational force to 1/4; tripling the distance reduces it to 1/9; increasing distance 10 times reduces force to 1/100. This geometric behaviour arises because gravity radiates from a point source spherically in three-dimensional space. Surface area of a sphere = 4πr², so the gravitational "intensity" spreads over ever-larger surface areas as distance grows. The same inverse-square law governs: electric force (Coulomb's law), light intensity, sound intensity, and any fundamental force or emitted energy from a point source in 3D space.
What is gravitational potential energy?
Gravitational potential energy U = −GMm/r is the energy stored in the gravitational interaction between two masses. The negative sign reflects that the system is bound — you must do work to separate the masses. As r → ∞, U → 0. At Earth's surface, U ≈ −62.5 MJ/kg (per kg of the object, using Earth's M and R). To escape, an object's kinetic energy must equal |U|: ½mv² ≥ GMm/r → v ≥ √(2GM/r). This is precisely the relation that defines escape velocity. Gravitational potential energy is why objects fall faster as they approach a massive body — they convert potential energy into kinetic energy.
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