Formulas & tables/Physics/Gravitation/First cosmic velocity (circular orbital speed)

First cosmic velocity (circular orbital speed)

v_I=\sqrt{\dfrac{GM}{R}}

Result unit: m/s (often given in km/s)

What it describes

The first cosmic velocity is the smallest speed that must be given to a body right at a planet's surface (or that of another celestial body) for it to enter a circular orbit around it — to become a satellite — instead of falling back to the surface.

Symbols

v_I

first cosmic velocity

[m/s]

G

gravitational constant (

\approx 6.67\cdot10^{-11}

)

[N·m²/kg²]

M

mass of the central body (e.g. the planet)

[kg]

R

orbital radius (at the surface ≈ the planet's radius)

[m]

Where it comes from

In a circular orbit gravity plays the role of the centripetal force. Setting them equal:

\dfrac{GMm}{R^2} = \dfrac{mv^2}{R}

. The satellite's mass

m

cancels, leaving

v^2 = \dfrac{GM}{R}

, that is

v_I = \sqrt{\dfrac{GM}{R}}

. Conclusion: the first cosmic velocity does NOT depend on the satellite's mass.

When to use it

For motion in a circular orbit just above the surface (

R \approx

the planet's radius). For Earth

v_I \approx 7.9

km/s (≈ 7900 m/s). Equivalently

v_I = \sqrt{gR}

, since

g = \dfrac{GM}{R^2}

Example

Find the first cosmic velocity for Earth.

Substitution

M = 5.97\cdot10^{24}

kg,

R = 6.37\cdot10^{6}

G = 6.67\cdot10^{-11}

Result

v_I = \sqrt{\dfrac{6.67\cdot10^{-11}\cdot 5.97\cdot10^{24}}{6.37\cdot10^{6}}} \approx 7.9\cdot10^{3}

m/s ≈ 7.9 km/s.

Related formulas

Second cosmic velocity (escape velocity) Newton's law of universal gravitation Gravitational acceleration

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