Relative speed of rocket depends on mass and impulse

The Tsiolkovsky formula determines the speed that an aircraft develops under the influence of the thrust of a rocket engine, unchanged in direction, in the absence of all other forces. For a rocket flying at a speed close to the speed of light, the generalized Tsiolkovsky formula is valid, in which the speed of light is present.

Notation:

  1. \(c\) (c) is speed_of_light.

speed

Final speed of the rocket

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