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:

$$\frac{m_1}{m_0}={\left(\frac{1-\frac{v}{c}}{1+\frac{v}{c}}\right)}^{\frac{c}{2v_{\text{e}}}}$$