Tsiolkovskii rocket equation 치올콥스키 로켓 방정식

EPFL Class

First, if velocity of exhaust gas is constant, \(\vec{v}_{e}=\rm{const}\)

$$\vec{F}=\frac{d\vec{P}}{dt}=\frac{d(m\vec{v}_{e})}{dt}=m\frac{d\vec{v}_{e}}{dt}+\vec{v}_{e}\frac{dm}{dt}=\dot{m}\vec{v}_{e}$$

$$\vec{F}=\dot{m}\vec{v}_{e}$$

 

Second, for velocity of the spacecraft.

$$\frac{d\vec{v}}{dt}=\frac{\vec{F}}{m}=\frac{\dot{m}\vec{v}_{e}}{m}=\vec{v}_{e}\frac{\dot{m}}{m}\cdots(1)$$

 

Third, integral equation (1) we derived above.

$$\Delta \vec{v}=\vec{v}_{e}\ln{\frac{m_{i}}{m_{f}}}$$

 

Also, we can define \(\vec{v}_{e}\) like below

$$\vec{v}_{e}=g_{0}I_{sp}$$

 

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