Example 1
v₂ ≈ 11.2 km/s.
Second Cosmic Velocity Calculator
Enter the mass and the distance from the center. You get the minimum escape speed. Earth at r = 6371 km is 11.19 km/s, which is v₁√2 at v₁ = 7.91 km/s.
Circular orbit: first cosmic velocity (v₂ = v₁√2). Field: g = GM/r². Circular path: a = v²/r. Force: F = GMm/r².
Result
Constant G (N·m²/kg², optional), Mass M (kg) and Distance r (m). The result shows up here.
How it works
Second cosmic velocity is the threshold where mechanical energy stops being negative. From ½mv² − GMm/r ≥ 0 you get v₂ = √(2GM/r). That is exactly v₁√2, because v₁ = √(GM/r). √2 ≈ 1.414, so escape is 41% faster than the circle.
At Earth’s surface, the same M and r as for v₁: 5.972×10²⁴ kg and 6.371×10⁶ m. 2GM/r = 1.251×10⁸, the square root is 11186 m/s, or 11.19 km/s. School 11.2 km/s is a rounding. The path at exactly v₂ is a parabola.
Higher up the threshold falls. On LEO r = 6.771×10⁶ m, v₂ ≈ 10.85 km/s. At geosynchronous r ≈ 4.216×10⁷ m you get about 4.35 km/s. A launch from Earth still has to climb through the atmosphere and the well on the way.
Projectile mass cancels. The formula skips direction: it gives a magnitude. A real launch uses Earth’s rotation and avoids thick air. Below v₂ the body comes back on an ellipse or falls, unless it has thrust.
M and r must be positive. Empty G = 6.67430×10⁻¹¹. The result note reminds you that v₂ = v₁√2 and also prints v₁ for the same inputs. The result is in m/s.
Moon at the surface: v₂ ≈ 2.38 km/s. Mars about 5.03 km/s. From the Sun-Earth scale (r = 1 au) solar escape is around 42 km/s. The circular speed v₁ is a separate page.
How to use
- Leave G empty, or type the constant from the problem.
- Type M in kilograms, for example 5.972e24.
- Type r from the center in meters, for example 6371000.
- Click Calculate. Earth and 6371 km give 11.19 km/s. Check that this is 7.91 × √2.
- The circle itself is next door as v₁ = √(GM/r). Here you stay with escape.
Formula
v2 = √(2GM / r)
Link to circular orbit: v2 = v1√2 with v1 = √(GM/r).
Letters in v₂ = √(2GM/r)
Escape threshold: v₂ = √(2GM/r) = v₁√2. The same M and r that gave 7.91 km/s now give 11.19 km/s, school 11.2.
- v₂
- Minimum escape speed. 2GM/r = 1.251×10⁸, square root 11186 m/s, or 11.19 km/s.
- G
- Again 6.67430×10⁻¹¹ if you leave it blank. The 2 under the root is what separates v₂ from circular v₁.
- M
- Mass 5.972×10²⁴ kg. √2 ≈ 1.414, so escape is 41% faster than 7.91 km/s.
- r
- Distance from the center, 6.371×10⁶ m at the surface. That is the escape start, not the ISS radius with v₁ = 7.67 km/s.
Real-life examples
Example 2
r ≈ 6771000 m.
Example 3
M ≈ 7.35e+22 kg, r ≈ 1737000 m.
Example 4
M ≈ 6.39e+23 kg, r ≈ 3390000 m.
Example 5
r = 12742000 m.
Example 6
r ≈ 42164000 m.
Example 7
escape from the Sun-Earth system (scale).
Example 8
M ≈ 1.9e+27 kg, r ≈ 69911000 m.
Example 9
G = 6.67e-11, Earth surface.
Example 10
r = 6800000 m.
Ways to use this calculator
- A quick v₂ for Earth and other bodies.
- A check of v₂ = v₁√2 on the same M and r.
Frequently asked questions
How much v₂ for Earth at r = 6371 km?
11186 m/s, or 11.19 km/s. School work often writes 11.2 km/s.
Why is v₂ = v₁√2?
Because v₁² = GM/r and v₂² = 2GM/r. The ratio is √2 ≈ 1.414. From 7.91 km/s you get 11.19 km/s.
How is escape different from orbit?
v₁ holds a circle (negative energy). v₂ lets you leave to infinity (zero energy). Above v₂ the path is a hyperbola.
Does v₂ depend on projectile mass?
No. The test mass cancels, as it does for v₁ and for g.
How much v₂ on LEO, r = 6771 km?
About 10.85 km/s. The farther you start, the lower the threshold.
Does direction matter?
The formula gives a magnitude. A real launch uses Earth’s rotation and the atmosphere.
What v₂ does the Moon have?
About 2.38 km/s at r ≈ 1737 km. Mars at the surface is about 5.03 km/s.
What path do you get at exactly v₂?
A parabola, zero energy. Above: hyperbola. Below: a bound orbit or a fall.
Can I change G?
Yes. Type G or leave it empty for 6.67430×10⁻¹¹. Escape speed still uses √(2GM/r).
Where do I compute the circle itself?
On the first cosmic velocity page. Same M and r, no 2 under the square root.
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.