Gravitational Velocity Calculator

Calculator inputs

Calculation type *

Choose a model before entering distances.

Celestial body preset

Presets fill μ and radius automatically.

Input method *

μ = G × M is computed when needed.

Gravitational parameter μ *

For Earth: μ ≈ 3.986004418e14 m³/s².

Body mass M *

G = 6.67430×10⁻¹¹ m³/(kg·s²).

Body radius R *

Required when using altitude mode.

Distance input style

r = R + h when altitude mode is selected.

Altitude h (above surface) *

For surface calculations, keep h = 0.

Distance r (from center) *

This bypasses R and altitude entirely.

Initial altitude h0 *

Starting point (higher altitude).

Final altitude h1 *

Ending point (lower altitude).

Initial distance r0 *

Distance from the body center.

Final distance r1 *

Must be smaller than r0.

Reset

Formula used

Escape velocity

Speed needed to reach infinity with zero remaining speed.

v_esc = √(2μ / r)

Where μ = GM and r is distance from center.

Circular orbital velocity

Speed for a perfectly circular orbit at radius r.

v_orb = √(μ / r)

Optional period: T = 2π √(r³ / μ).

Free-fall velocity (from rest)

From initial distance r0 down to r1 without drag.

v = √(2μ(1/r1 − 1/r0))

Use altitude mode with r = R + h.

How to use this calculator

Select the calculation type: escape, orbital, or free-fall.
Pick a body preset, or keep Custom.
Choose μ directly or mass + G.
Select distance style: altitude above surface or center distance.
Enter required values, then press Calculate.
Use the CSV/PDF buttons to export your results.

Example data table

Approximate values at each body's surface (h = 0).

Body	μ (m³/s²)	Radius R (m)	Escape velocity (km/s)	Orbital velocity (km/s)
Earth	3.986004418e14	6,371,000	11.186	7.910
Moon	4.9048695e12	1,737,400	2.376	1.680
Mars	4.282837e13	3,389,500	5.027	3.555
Jupiter	1.26686534e17	69,911,000	60.202	42.569

Gravitational velocity guide

1) What “gravitational velocity” means

In orbital mechanics, “gravitational velocity” usually refers to the speed set by gravity at a given distance from a body’s center. This calculator provides three useful speeds: escape velocity, circular orbital velocity, and free-fall velocity between two radii.

2) The key constant: μ (standard gravitational parameter)

Most space calculations use μ = GM instead of mass alone. It bundles the gravitational constant and the body’s mass into a single value with units of m³/s². For Earth, μ is about 3.986×10¹⁴ m³/s², while the Moon is roughly 4.905×10¹² m³/s².

3) Distance matters more than you think

Speeds depend on r, the distance from the center of the body. If you raise altitude, r increases, so required orbital speed drops. For example, near Earth’s surface (r ≈ 6,371 km), circular speed is about 7.91 km/s, while at 400 km altitude it is slightly lower.

4) Escape velocity data points

Escape velocity comes from v = √(2μ/r). At Earth’s surface it is about 11.19 km/s; on the Moon it is about 2.38 km/s. Jupiter’s deep gravity gives a much larger surface escape speed, around 60 km/s using its mean radius.

5) Circular orbit speed and period

Circular orbital speed is v = √(μ/r). The calculator also reports the circular period T = 2π√(r³/μ). Around Earth, a low orbit (hundreds of km up) typically yields a period near 90 minutes, which is why many satellites circle Earth multiple times per day.

6) Free-fall velocity between two radii

Free-fall speed here assumes the object starts from rest and air drag is ignored. It uses v = √(2μ(1/r1 − 1/r0)). This is an energy method, so it is helpful for comparing “drop” speeds from high altitude to a lower altitude.

7) Using altitude vs. center distance

Altitude mode is convenient when your input is “above the surface.” The calculator converts altitude to center distance by r = R + h. Center-distance mode is best for astronomy problems where r is already measured from the body’s center.

8) Practical accuracy notes

These formulas assume a spherical body, no atmosphere, and a two-body system. Real missions include rotation, non-spherical gravity, thrust, and drag. Still, these results are excellent for quick estimates, unit checks, and building intuition before using detailed simulators.

FAQs

1) What is μ and why is it better than mass?

μ equals G multiplied by mass. It avoids repeatedly multiplying by G and is often known more accurately for planets, making orbital and escape calculations simpler and more consistent.

2) Which speed should I use for a satellite?

Use circular orbital velocity for a circular orbit at a chosen altitude. For non-circular orbits, you would need vis‑viva and orbital elements, which are beyond this basic circular case.

3) Why does altitude mode require body radius?

Altitude is measured from the surface, but gravity formulas use distance from the center. Radius converts altitude to center distance using r = R + h.

4) Does escape velocity mean I instantly leave the planet?

No. It is the minimum speed at a given radius to reach infinity with zero remaining speed, ignoring drag and propulsion limits. Real launches require more due to atmosphere and trajectory losses.

5) Can free-fall velocity exceed orbital velocity?

Yes, depending on r0 and r1. Falling from far away toward a small r1 can produce high speeds. However, real bodies and atmospheres often limit actual speeds through heating and drag.

6) What units should I enter for μ?

Use m³/s² if you are entering meters and kilometers consistently. If you have μ in km³/s², select that unit; the calculator converts it internally to m³/s².

7) Are these values exact for Earth?

They are close but idealized. Earth’s rotation and non-spherical gravity slightly change real-world values. For most educational and quick engineering estimates, the differences are small.