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Universal Gravitation Calculator

Enter two masses and the distance between them to calculate gravitational force, field strength, orbital velocity, and escape velocity. Every result includes a step-by-step breakdown of the formula with your values substituted in.

Force Diagram

m₁5.972 × 10^24 kgm₂7.342 × 10^22 kgr = 3.844 × 10^8 mF = 1.98 × 10^20 N

Parameters

kg
kg
m

Results

Gravitational Force (F)
1.98 × 10^20 N
Field strength (g) at m₁
2.697 × 10^-3 m/s²
Field strength (g) at m₂
3.316 × 10^-5 m/s²
Orbital velocity around m₁
1018 m/s
Orbital velocity around m₂
112.9 m/s
Escape velocity from m₁
1440 m/s
Escape velocity from m₂
159.7 m/s

Step-by-Step Calculation

1. Gravitational Force

F=G⋅m1⋅m2r2F = \frac{G \cdot m_1 \cdot m_2}{r^2}
F=(6.674×10−11)(5.972×1024)(7.342×1022)(3.844×108)2F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})(7.342 \times 10^{22})}{(3.844 \times 10^{8})^2}
F=1.98×1020 NF = 1.98 \times 10^{20} \text{ N}

2. Field Strength due to m₁

g1=G⋅m1r2g_1 = \frac{G \cdot m_1}{r^2}
g1=(6.674×10−11)(5.972×1024)(3.844×108)2g_1 = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})}{(3.844 \times 10^{8})^2}
g1=2.697×10−3 m/s2g_1 = 2.697 \times 10^{-3} \text{ m/s}^2

3. Orbital Velocity around m₁

vorb=G⋅m1rv_{\text{orb}} = \sqrt{\frac{G \cdot m_1}{r}}
vorb=(6.674×10−11)(5.972×1024)3.844×108v_{\text{orb}} = \sqrt{\frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})}{3.844 \times 10^{8}}}
vorb=1018 m/sv_{\text{orb}} = 1018 \text{ m/s}

4. Escape Velocity from m₁

vesc=2G⋅m1rv_{\text{esc}} = \sqrt{\frac{2G \cdot m_1}{r}}
vesc=2(6.674×10−11)(5.972×1024)3.844×108v_{\text{esc}} = \sqrt{\frac{2(6.674 \times 10^{-11})(5.972 \times 10^{24})}{3.844 \times 10^{8}}}
vesc=1440 m/sv_{\text{esc}} = 1440 \text{ m/s}

Reference Guide

Newton's Law of Gravitation

Every object with mass attracts every other object with mass. The force depends on both masses and the square of the distance between them.

F=G⋅m1⋅m2r2F = \frac{G \cdot m_1 \cdot m_2}{r^2}

where G=6.674×10−11  N⋅m2/kg2G = 6.674 \times 10^{-11}\;\text{N}\cdot\text{m}^2/\text{kg}^2 is the gravitational constant. The force is always attractive and acts along the line connecting the two centers of mass.

Gravitational Field Strength

The gravitational field strength at a distance rr from a mass MM tells you the acceleration any object would experience at that point.

g=GMr2g = \frac{GM}{r^2}

On Earth's surface, g≈9.81  m/s2g \approx 9.81\;\text{m/s}^2. On the Moon it is about 1.62  m/s21.62\;\text{m/s}^2, roughly one sixth of Earth's value.

Orbital Velocity

For a circular orbit, gravitational force provides the centripetal acceleration. Setting them equal gives the speed needed to maintain the orbit.

vorb=GMrv_{\text{orb}} = \sqrt{\frac{GM}{r}}

A satellite in low Earth orbit (about 400 km altitude) needs roughly 7,670 m/s. Higher orbits require less speed but take longer to complete one revolution.

Escape Velocity

Escape velocity is the minimum speed needed to leave a gravitational field without further propulsion. It comes from setting kinetic energy equal to gravitational potential energy.

vesc=2GMrv_{\text{esc}} = \sqrt{\frac{2GM}{r}}

Notice that escape velocity is exactly 2\sqrt{2} times the orbital velocity at the same distance. For Earth's surface, it is about 11,200 m/s.