Tsiolkovskii rocket equation 치올콥스키 로켓 방정식

EPFL Class

First, if velocity of exhaust gas is constant, ${\overrightarrow{v}}_e=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$

$$\overrightarrow{F}=\frac{d\overrightarrow{P}}{dt}=\frac{d(m{\overrightarrow{v}}_e)}{dt}=m\frac{d{\overrightarrow{v}}_e}{dt}+{\overrightarrow{v}}_e\frac{dm}{dt}=\dot{m}{\overrightarrow{v}}_e$$

$$\overrightarrow{F}=\dot{m}{\overrightarrow{v}}_e$$

 

Second, for velocity of the spacecraft.

$$\frac{d\overrightarrow{v}}{dt}=\frac{\overrightarrow{F}}{m}=\frac{\dot{m}{\overrightarrow{v}}_e}{m}={\overrightarrow{v}}_e\frac{\dot{m}}{m}\cdots (1)$$

 

Third, integral equation (1) we derived above.

$$\mathrm{\Delta }\overrightarrow{v}={\overrightarrow{v}}_e\ln \frac{m_i}{m_f}$$

 

Also, we can define ${\overrightarrow{v}}_e$ like below

$${\overrightarrow{v}}_e=g_0{I}_{sp}$$

 

Here, $I_{sp}$ is specific Impulse, in Korean '비추력' with unit of 'second(s)'