Tsiolkovsky rocket equation

A Tsiolkovsky rocket equation describes the motion of an idealized rocket. It is named after the Russian researcher Konstantin Eduardovich Tsiolkovsky, who extensively studied space travel and laid the scientific foundations for the theory of multistage rockets.

The Tsiolkovsky rocket equation applies to the idealized case with no gravity and no air resistance (vacuum):

$$v(t) = v_g \cdot \ln \left(\frac{m_0}{m(t)}\right),$$

• $v(t)$ is the rocket's velocity at time $t$,
• $v_g$ is the speed of the exhaust gas relative to the rocket,
• $m_0$ is the rocket's initial mass (often simply denoted as $m$),
• $m(t)$ is the rocket's mass after time $t$ has elapsed from launch, sometimes denoted as capital $M$.

With minor rearrangements, the equation in a "simplified" form is:

$$\Delta v = v \cdot \ln \left(\frac{m}{M}\right)$$