何もない空間でのロケット推進 - I

Rocket propulsion exemplifies Newton's third law: fuel combustion ejects gases at high velocity in one direction, creating an equal and opposite force that propels the rocket forward. In space, without friction, this action-reaction pair is the sole mechanism driving motion. The expelled gases push backward while the rocket accelerates forward, demonstrating the fundamental principle that forces always occur in equal and opposite pairs.

In empty space, no external forces act on the rocket-fuel system, making it isolated. Therefore, total momentum remains constant: initial momentum of the loaded rocket equals final momentum of the rocket plus expelled gases combined. As fuel combusts and ejects, the system's total momentum is conserved, though individual components' momenta change. This principle allows calculation of rocket velocity and acceleration at any instant during flight.

As the rocket ejects burned fuel gases, its total mass continuously decreases while its velocity increases. The expelled gas mass travels opposite to the rocket's direction at high velocity. Due to momentum conservation, the rocket's momentum change equals the expelled gases' momentum change with opposite sign. This inverse relationship between decreasing mass and increasing velocity is fundamental to rocket dynamics in space.

Starting with initial momentum mv of the loaded rocket, fuel combustion over time dt produces expelled gas mass dmex moving at velocity vex in the negative direction. Applying momentum conservation and expressing the equation in terms of magnitude, smaller terms are neglected to obtain the simplified conservation of linear momentum equation for the rocket. This derivation connects microscopic fuel ejection to macroscopic rocket acceleration.

The ideal rocket equation calculates the rocket's velocity at any given instant using conservation of momentum principles. Beyond velocity prediction, this equation enables estimation of thrust acting on the rocket and its instantaneous acceleration. These calculations depend on the expelled gas mass, ejection velocity, and remaining fuel mass, making the ideal rocket equation essential for analyzing rocket dynamics in empty space.

In empty space, no medium exists to create friction forces. Unlike vehicles on Earth where friction drives motion, rockets rely entirely on momentum exchange between expelled gases and the rocket body. The absence of friction means the rocket's acceleration depends solely on fuel ejection velocity and mass rate, making momentum conservation the governing principle for all motion changes.

Expelled gases traveling in the negative x-direction at high velocity create momentum in that direction. By momentum conservation, the rocket gains equal momentum in the positive x-direction, accelerating forward. Higher ejection velocity and greater expelled mass increase the momentum transfer, producing greater rocket acceleration. The direction opposite to gas ejection determines the rocket's acceleration direction.