Earth surface
Earth M and R: v₁ ≈ 7.9 km/s.
First Cosmic Velocity Calculator
Enter G (optional), central mass M, and orbit radius r. You get v₁ for a circular orbit. At Earth surface that is about 7.9 km/s.
In orbit a = v₁²/r = GM/r²: see centripetal acceleration and g. Force: centripetal force = gravitational force. Escape: second cosmic velocity (v₂ = v₁√2).
Result
Enter values. The result shows up here.
How it works
First cosmic velocity v₁ is the speed on a circular orbit around mass M. Balance: gravity supplies centripetal acceleration, so GM/r² = v₁²/r, hence v₁ = √(GM/r). Prefer separate G, M, and r over a bare GM product: you see where the result comes from.
At Earth surface (r ≈ 6371000 m) you get v₁ ≈ 7900 m/s (about 7.9 km/s). That is a “surface” value without atmosphere. On LEO (larger r) v₁ is a bit smaller, because r sits under the square root.
Same link as a = v²/r on centripetal acceleration: in orbit a = g = GM/r². The inward force is gravity, also written as F = m·v²/r.
M and r must be positive. Blank G = 6.67430×10⁻¹¹. We show the result in m/s (and km/h or mph from the header). Escape needs v₂ = v₁√2.
How to use
- Optionally enter G (blank = 6.67430×10⁻¹¹).
- Enter central mass M (> 0).
- Enter orbit radius r (> 0).
- Read v₁ = √(GM/r). Chart: v₁ vs r at fixed M.
Formula
v1 = √(GM / r)
Equivalently v12 / r = GM / r2 (centripetal a = g).
Real-life examples
LEO 400 km
r ≈ 6771000 m.
LEO 500 km
r ≈ 6871000 m.
Low lunar orbit
M ≈ 7.35e+22 kg, r ≈ 1837000 m.
Mars surface (scale)
M ≈ 6.39e+23 kg, r ≈ 3390000 m.
Geostationary
r ≈ 42164000 m.
2× Earth R
r = 12742000 m.
Sun at 1 au
Earth orbit around the Sun (scale).
Custom G
G = 6.67e-11, Earth surface.
ISS r = 6800 km
r = 6800000 m.
Ways to use this calculator
- Quick v₁ for Earth, Moon, or Mars.
- Compare surface v₁ with LEO v₁.
Frequently asked questions
Why not only the GM product?
You can, but separate G, M, and r shows the model better and lets you change G. Richer UX than a thin GM-only page.
How does this link to centripetal a?
In orbit a = v₁²/r = GM/r². See <a href="centripetal-acceleration.html">centripetal acceleration</a>.
What is v₁ at Earth?
About 7.9 km/s at r = R_E. The exact value depends on the M and R you adopt.
Can you fly that low above the surface?
The formula says yes; atmosphere and terrain say no. Real orbits are higher.
How do I get v₂?
v₂ = v₁√2 on <a href="second-cosmic-velocity.html">second cosmic velocity</a>.
Does v₁ depend on satellite mass?
No. Satellite mass cancels; you keep √(GM/r).
How does this connect to force?
F = GMm/r² = m·v₁²/r. See <a href="gravitational-force.html">gravitational force</a> and <a href="centripetal-force.html">centripetal force</a>.
Why are r ≤ 0 or M ≤ 0 rejected?
Radius and mass must be positive.
What about energy?
On a circular orbit E = −GMm/(2r). Near the surface compare with <a href="potential-energy.html">potential energy</a>.
Does the v₁(r) chart fall?
Yes: at fixed M, larger r means smaller v₁.
What units does the result use?
m/s (plus km/h / mph from the switch). G stays in SI.