How Gravitational Acceleration is Calculated
The acceleration due to gravity at the surface of a planetary body is derived from Newton's law of universal gravitation. The formula is:
g = GM / r²
Where G is the universal gravitational constant (6.67430 x 10−11 N m²/kg²), M is the mass of the body in kilograms, and r is the radius of the body in metres. A more massive body produces stronger surface gravity, while a larger radius reduces it because the surface is further from the centre of mass.
Free-Fall Kinematics
Once you know g, you can calculate the motion of an object falling from rest using the standard kinematic equations (assuming no air resistance):
- v = gt (final velocity after falling for time t)
- d = ½gt² (distance fallen in time t)
- v² = 2gd (velocity after falling distance d)
- t = v/g (time to reach velocity v)
- t = √(2d/g) (time to fall distance d)
These equations apply equally on any body with known g, which is why astronauts on the Moon fall in slow motion and objects on Jupiter fall with crushing speed.
Worked Example
An object is dropped from rest on Earth and falls for 3 seconds. Using G = 6.67430 x 10−11 N m²/kg², M = 5.9722 x 10²&sup4; kg, r = 6,371,000 m:
- g = GM/r² = (6.67430 x 10−11 x 5.9722 x 10²&sup4;) / (6,371,000)² = 9.820 m/s²
- Final velocity: v = gt = 9.820 x 3 = 29.46 m/s (106.1 km/h)
- Distance fallen: d = ½gt² = 0.5 x 9.820 x 9 = 44.19 m
This matches the calculator's default output with body set to Earth and fall time set to 3 seconds. The value differs slightly from the standard gravity (9.807 m/s²) because that figure accounts for Earth's rotation; the pure Newtonian formula GM/r² at the mean surface radius gives approximately 9.820 m/s².
Why g Varies by Location on Earth
The standard value of g = 9.807 m/s² is an internationally defined standard. In reality, g varies slightly across Earth's surface because:
- Earth is not a perfect sphere: it bulges at the equator, so the equatorial surface is further from the centre (g is slightly lower at the equator, approximately 9.780 m/s²)
- Earth's rotation produces a centrifugal effect that reduces the apparent gravity at the equator
- Altitude reduces g: every 1,000 m gain in altitude reduces g by about 0.003 m/s²
- Local geology and density variations cause minor anomalies
For most physics calculations, 9.8 m/s² or 9.81 m/s² is used. The precise value 9.807 m/s² is the standard adopted by NIST and the International Bureau of Weights and Measures.
Gravity on Other Planets
Surface gravity varies dramatically across the solar system. Standing on Jupiter, you would weigh about 2.6 times your Earth weight. On the Moon you would weigh roughly one sixth as much, and on Mars about 38% of your Earth weight. These differences have profound implications for human spaceflight, structural engineering, and the physics of planetary atmospheres.
Related Calculators
Sources and method: Gravitational constant G = 6.67430 x 10−11 N m²/kg² from NIST CODATA 2022. Planetary mass and radius values from NASA Solar System Exploration planetary fact sheets. Standard gravity 9.80665 m/s² from ISO 80000-3. Free-fall kinematic equations from standard Newtonian mechanics.
All calculations assume an object falling from rest in a vacuum (no air resistance) at the surface of the selected body. Actual free-fall on Earth is affected by air resistance, which becomes significant above approximately 5 m/s for a human body. For precise scientific work, consult specialist references.
