Formula: Second Cosmic Velocity Second cosmic velocity Radius of the celestial body Mass Gravitational constant
$$\class{purple}{v_2} ~=~ \sqrt{\frac{2G\, \class{brown}{M_{\text p}} }{R_{\text p}}}$$
$$\class{purple}{v_2} ~=~ \sqrt{\frac{2G\, \class{brown}{M_{\text p}} }{R_{\text p}}}$$
$$R_{\text p} ~=~ \frac{ 2 G\, \class{brown}{M_{\text p}} }{ {\class{purple}{v_2}}^2 }$$
$$\class{brown}{M_{\text p}} ~=~ \frac{ R_{\text p} \, {\class{purple}{v_2}}^2 }{ 2G }$$
$$G ~=~ \frac{ R{\text p} \, {\class{purple}{v_2}}^2 }{ 2\class{brown}{M_{\text p}} }$$
Second cosmic velocity
$$ \class{purple}{v_2} $$ Unit $$ \frac{\mathrm m}{\mathrm s} $$
Second cosmic velocity is the velocity necessary to escape without propulsion from the gravitational field of a celestial body.
For a rocket to escape the Earth's gravitational field, it must have the following minimum velocity: \[ \class{purple}{v_2} ~=~ \sqrt{ 2 ~\cdot~ \frac{6.67 \cdot 10^{-11} \frac{\mathrm N \, \mathrm{m}^2}{\mathrm{kg}^2} ~\cdot~5.97 \cdot 10^{24}\,\mathrm{kg} }{6.38 \cdot 10^6 \,\mathrm{m}} } ~=~ 11.2 \, \frac{\mathrm{km}}{\mathrm s} \]
Radius of the celestial body
$$ R_{\text p} $$ Unit $$ \mathrm{m} $$
Radius of the celestial body you are trying to escape. For example, radius of the Earth.
Mass
$$ \class{brown}{M_{\text p}} $$ Unit $$ \mathrm{kg} $$
Mass of the celestial body. In the case of the Earth, the mass is: \( 5.972 \cdot 10^{24} \, \mathrm{kg} \).
Gravitational constant
$$ G $$ Unit $$ \frac{\mathrm{N} \, \mathrm{m}^2}{\mathrm{kg}^2} = \frac{\mathrm{m}^3}{\mathrm{kg} \, \mathrm{s}^2} $$
The gravitational constant is a physical constant that occurs in equations describing the interaction between masses. It has the following experimentally determined value:
$$ G ~\approx~ 6.674 \, 30 ~\cdot~ 10^{-11} \, \frac{ \mathrm{m}^3 }{ \mathrm{kg} \, \mathrm{s}^2 } $$