Cosmic velocities (I and II)

What just happened?

You changed the mass of the celestial body — both velocities grew, but the escape velocity stayed exactly $\sqrt{2}$ times larger than the orbital one. This ratio does not depend on the planet.

Cosmic velocities:

$v_I$ — first cosmic velocity: the minimum speed to stay in a circular orbit just above the surface. $v_{II}$ — second cosmic velocity: the minimum speed to escape the gravitational field entirely. G — gravitational constant, M — mass of the body, R — its radius.

Why do they depend on M and R?

A larger mass M pulls harder → you must travel faster. A larger radius R moves you farther from the centre, weaker gravity at the surface → the velocities drop. For Earth $v_I\approx 7.9$ km/s, and $v_{II}\approx 11.2$ km/s.

Why is it easier to lift off from the Moon?

The Moon has a small mass and radius, so both velocities are several times smaller than on Earth ($v_{II}\approx 2.4$ km/s). That is why the lander needed far less fuel to launch back up. It is the same physics as universal gravitation.

Why you need this for the exam:

The formulas $v_I=\sqrt{GM/R}$ and $v_{II}=v_I\sqrt{2}$ solve most problems about satellites and escape. The trick: compute $v_I$, and you get the second velocity by multiplying by $\sqrt{2}$.

Remember: the escape velocity is always $\sqrt{2}$ times larger than the orbital one — no matter which planet you launch from.