Variation of g with Height: Formula & Explanation

This calculator computes the acceleration due to gravity at a given height above Earth’s surface. You enter the height, and it returns the value of g at that altitude, using the standard inverse-square formula. The result is shown in meters per second squared (m/s²).

Formula

The acceleration due to gravity at a height h above the Earth’s surface is given by:

g_h = g₀ × (R / (R + h))²

Where:

• g_h = gravity at height h (m/s²)
• g₀ = gravity at sea level = 9.80665 m/s² (standard value)
• R = mean Earth radius = 6371 km (or 6,371,000 m)
• h = height above the surface (same unit as R, so convert km to m if needed)

This is a direct consequence of Newton’s law of universal gravitation: the force between two masses is inversely proportional to the square of the distance between their centers. As you move away from Earth, the distance from the center increases from R to R+h, so gravity weakens.

Worked Example

Using the calculator’s default height of 10 km:

1. Convert height to meters: 10 km = 10,000 m (but since R is in km, we can keep h in km: 10 km).
2. Add to Earth’s radius: R + h = 6371 + 10 = 6381 km.
3. Ratio: R / (R + h) = 6371 / 6381 ≈ 0.99843.
4. Square: (0.99843)² ≈ 0.99686.
5. Multiply by g₀: 9.80665 × 0.99686 ≈ 9.7758 m/s².

The calculator shows 9.7758 m/s², matching this calculation. The difference from sea level is about 0.0309 m/s² – a small but measurable drop.

Why Gravity Changes with Height

Gravity is not constant everywhere. It depends on two main factors: the mass of the Earth and the distance from its center. The inverse-square law means that if you double the distance from the center, gravity becomes one-fourth as strong. But because Earth’s radius is about 6,371 km, even a height of 10 km (0.16% of the radius) only reduces gravity by about 0.3%.

For most everyday purposes – like engineering buildings or calculating weight on a mountain – this small change is negligible. However, for satellite orbits, high-altitude physics, or precise geophysical measurements, the variation matters.

How to Use This Calculator

Enter the height in kilometers (or meters – the calculator accepts any positive number). The first output is the gravity at that height. The second output shows how much weaker it is compared to sea level. This helps you quickly see the magnitude of the change.

For example:

• At the top of Mount Everest (about 8.8 km), gravity is roughly 9.792 m/s².
• On a commercial flight (about 10 km), it’s about 9.776 m/s².
• At the International Space Station orbit (about 400 km), it’s about 8.68 m/s² – still 88% of surface gravity!

Common Mistakes and Misconceptions

A frequent error is mixing units. If you use R in kilometers, h must also be in kilometers. The calculator handles this internally, but if you do it by hand, always convert to the same unit.

Another misconception: astronauts in orbit are not weightless because gravity is zero. At 400 km, gravity is still about 88% of surface value. They feel weightless because they are in free fall – constantly falling toward Earth but moving sideways fast enough to miss it.

Also, the formula assumes a perfectly spherical Earth of uniform density. In reality, Earth is an oblate spheroid (bulges at the equator), and gravity varies with latitude and local geology. The formula gives a good approximation for most purposes but not exact values for specific locations.

Limits of the Formula

This inverse-square formula works well for heights above the surface, but it has limits:

• It ignores Earth’s rotation (which causes a slight centrifugal effect, reducing apparent gravity at the equator).
• It assumes a uniform spherical Earth, not the actual ellipsoid shape.
• It doesn’t account for local mass variations (mountains, ocean trenches, underground density).
• For very large heights (comparable to Earth’s radius), the formula is still accurate, but you must use a consistent value for R.

For precise work, geodesists use more complex models like the International Gravity Formula, which includes latitude and altitude corrections. But for educational and general engineering purposes, the simple inverse-square law is sufficient.

Frequently Asked Questions

Does gravity decrease linearly with height?

No. Gravity decreases with the square of the distance from the center, not linearly. So doubling the height does not halve gravity; it reduces it to one-quarter (if you double the distance from the center). Near the surface, the change is roughly linear for small heights, but the exact formula is quadratic.

At what height is gravity half of Earth’s surface value?

Set g_h = 0.5 × g₀. Then (R/(R+h))² = 0.5, so R/(R+h) = √0.5 ≈ 0.7071. Solving for h gives h = R(1/0.7071 – 1) ≈ 0.414R ≈ 2,637 km. So gravity halves at about 2,637 km above the surface.

Why do astronauts float if gravity is still strong at their altitude?

Astronauts are in free fall. The spacecraft and everything in it are accelerating toward Earth at the same rate, so there is no normal force pushing against them. This creates the sensation of weightlessness, even though gravitational pull is still significant.

Does the formula work for other planets?

Yes, if you replace g₀ and R with the values for that planet. For example, on Mars, g₀ ≈ 3.71 m/s² and R ≈ 3,390 km. The same inverse-square relationship applies.

How much does gravity change at the top of a skyscraper?

For the Burj Khalifa (828 m), the change is about 0.00026 m/s² – roughly 0.003% of surface gravity. That’s far too small to notice in daily life, but precise instruments can detect it.

What is the difference between g and G?

Lowercase g is the acceleration due to gravity (about 9.8 m/s² on Earth). Uppercase G is the universal gravitational constant (6.674 × 10⁻¹¹ N·m²/kg²), which appears in Newton’s law and is the same everywhere in the universe.

Related

• How to Calculate h-Index in Research
• Multiply Out Brackets With Fractions: Step-by-Step
• Multiplying Non-Unit Fractions by an Integer

in h | Height above surface | km | 10 | 0
out g_h | Gravity at height | m/s² | 9.80665 * (6371 / (6371 + h))^2 | 4
out g_diff | Difference from sea level | m/s² | 9.80665 - 9.80665 * (6371 / (6371 + h))^2 | 4