Example 1
Earth M and R: v₁ ≈ 7.9 km/s.
First Cosmic Velocity Calculator
Enter the planet mass and the orbit radius. You get the circular-orbit speed. Earth at r = 6371 km is 7.91 km/s. Higher, at r = 6771 km, v₁ falls to 7.67 km/s.
In orbit a = v₁²/r = GM/r²: centripetal acceleration and g. Force: centripetal = gravity. Escape: v₂ = v₁√2.
Result
Constant G (N·m²/kg², optional), Mass M (kg) and Orbit radius r (m). The result shows up here.
How it works
First cosmic velocity is the speed on a circular orbit. Gravity supplies the inward acceleration, so GM/r² = v₁²/r. That is v₁ = √(GM/r). Typing G, M, and r separately shows where the number comes from, better than a single GM product.
At Earth’s surface r = 6.371×10⁶ m and M = 5.972×10²⁴ kg. GM/r = 6.256×10⁷, the square root is 7910 m/s, or 7.91 km/s. School 7.9 km/s is a rounding. The formula has no atmosphere: you still cannot fly just above the ground.
On LEO r = 6.771×10⁶ m, v₁ = 7670 m/s, about 7.67 km/s. Larger r in the denominator under the root means a smaller speed. At geosynchronous r ≈ 4.216×10⁷ m, v₁ falls to about 3.07 km/s.
Satellite mass cancels. The same v₁ for 1 kg and for a station. The inward force is gravity, also written F = m v₁²/r. Escape from the well is v₂ = v₁√2, about 11.19 km/s at the same surface.
M and r must be positive. Empty G = 6.67430×10⁻¹¹. The result is in m/s. The calculator also shows km/h or mph from the header. A comma and a period in 5.972e24 both parse.
The chart of v₁(r) at fixed M falls. Twice the r cuts v₁ by √2, not by half. Escape speed and g itself live on the neighboring pages, with the same M and r.
How to use
- Leave G empty, or type the constant from the problem.
- Type M in kilograms, for example 5.972e24.
- Type orbit radius r in meters: 6371000 at the surface, 6771000 on LEO.
- Click Calculate. Earth and 6371 km give 7.91 km/s. The same M and 6771 km give 7.67 km/s.
- Escape is on the neighbouring page as v₂ = √(2GM/r). Here you stay on the circle.
Formula
v1 = √(GM / r)
Equivalently v12 / r = GM / r2 (centripetal a = g).
Letters in v₁ = √(GM/r)
Circular orbit: v₁ = √(GM/r). Earth at r = 6371 km is 7.91 km/s; higher, at r = 6771 km, v₁ falls to 7.67 km/s.
- v₁
- Circular speed in m/s. GM/r = 6.256×10⁷, square root 7910 m/s, or 7.91 km/s at the surface.
- G
- Gravitational constant, optional. Together with M and r it builds v₁; blank G takes 6.67430×10⁻¹¹.
- M
- Planet mass, 5.972×10²⁴ kg for Earth. This is not v₂ = v₁√2 from the escape card.
- r
- Orbit radius in meters. 6.371×10⁶ m is 7.91 km/s; 6.771×10⁶ m is already 7.67 km/s.
Real-life examples
Example 2
r ≈ 6771000 m.
Example 3
r ≈ 6871000 m.
Example 4
M ≈ 7.35e+22 kg, r ≈ 1837000 m.
Example 5
M ≈ 6.39e+23 kg, r ≈ 3390000 m.
Example 6
r ≈ 42164000 m.
Example 7
r = 12742000 m.
Example 8
Earth orbit around the Sun (scale).
Example 9
G = 6.67e-11, Earth surface.
Example 10
r = 6800000 m.
Ways to use this calculator
- A quick v₁ for Earth, the Moon, or Mars.
- A comparison of 7.91 km/s at the surface with 7.67 km/s on LEO.
Frequently asked questions
How much v₁ for Earth at r = 6371 km?
7910 m/s, or 7.91 km/s. On LEO at 6771 km it is 7.67 km/s.
Does v₁ depend on satellite mass?
No. Satellite mass cancels. You are left with √(GM/r).
How does this connect to centripetal a?
In orbit a = v₁²/r = GM/r². That is the same g, written through speed.
Can you fly this way just above the surface?
The formula says yes. Atmosphere and mountains say no. Real orbits sit higher, hence 7.67 km/s on LEO, not 7.91.
How do I get v₂?
v₂ = v₁√2. From 7.91 km/s you get 11.19 km/s. A separate page computes √(2GM/r).
How much v₁ on a lunar orbit, r ≈ 1837 km?
M ≈ 7.35×10²² kg gives about 1.63 km/s. Mars at the surface is about 3.55 km/s.
Can I type GM alone?
The calculator needs G, M, and r separately. The product still happens inside. That way you can change G or M.
How much v₁ for Earth around the Sun?
M_☉ ≈ 1.989×10³⁰ kg, r ≈ 1.496×10¹¹ m, v₁ ≈ 29.8 km/s. Scale, not a precise ellipse.
Does the v₁(r) chart fall?
Yes. At fixed M, larger r gives smaller v₁, like 1/√r.
How does this link to force?
F = GMm/r² = m v₁²/r. Gravity and centripetal force are the same pair on a circle.
Knowledge sources
The formula is the school one. Units follow SI; NIST SP 330 and BIPM define the measures, not your result.
Page updated in 2026.