Relative rocket speed from mass change and effective exhaust speed¶

The Tsiolkovsky formula determines the speed that an aircraft develops due to the constant-direction thrust of the rocket engine in the absence of other forces. The generalized Tsiolkovsky formula is valid for a rocket flying at a speed close to the speed of light.

Notation:

\(c\) (c) is speed_of_light.

Links:

Wikipedia.

speed¶: Final speed of the rocket in the inertial reference frame where the rocket started at rest.

Symbol:: v
Latex:: \(v\)
Dimension:: velocity

effective_exhaust_speed¶: Effective exhaust speed of the rocket engine.

Symbol:: v_e
Latex:: \(v_\text{e}\)
Dimension:: velocity

initial_mass¶: Initial mass of the rocket.

Symbol:: m_0
Latex:: \(m_{0}\)
Dimension:: mass

final_mass¶: Final mass of the rocket.

Symbol:: m_1
Latex:: \(m_{1}\)
Dimension:: mass

law¶

m_1 / m_0 = ((1 - v / c) / (1 + v / c))^(c / (2 * v_e))

Latex:: \[\frac{m_{1}}{m_{0}} = \left(\frac{1 - \frac{v}{c}}{1 + \frac{v}{c}}\right)^{\frac{c}{2 v_\text{e}}}\]