Momentum in the dynamics of variable-mass systems: classical and relativistic case

TL;DR

This paper clarifies the correct formulation of momentum and equations of motion for variable-mass systems in classical and relativistic contexts, highlighting common misconceptions and providing a comprehensive perspective.

Contribution

It presents a detailed analysis of momentum in variable-mass systems, including relativistic extensions, and addresses ambiguities in the standard equations of motion.

Findings

The simple relation between momentum derivative and external force is frame-dependent.

Correct equations of motion for variable-mass systems are often misrepresented in textbooks.

Extension of classical equations to relativistic cases, including rocket motion, is demonstrated.

Abstract

We discuss a role of a momentum vector in the description of dynamics of systems with variable mass, and show some ambiguity in expressing the 2nd Newtonian law of dynamics in terms of momentum change in time for variable-mass systems. A simple expression that the time-derivative of the momentum of the body with variable mass is equal to the net external force is not always true (only if a special frame of reference is assumed). In basic textbooks and multiple lecture notes the correct equation of motion for a variable-mass system (including relative velocities of the masses entering or leaving the body) is not sufficiently well discussed, leading to some problems with understanding the dynamics of these systems among students. We also show how the equation of motion in classical case (in translational motion) can be easily expanded to the relativistic case and discuss a motion of a relativistic rocket. For the non-relativistic case also a rotational motion is discussed. It is of course true, that most of the good literature treats the problem accurately, but some of the commonly used textbooks do not. The purpose of our letter is to pay attention to the problem of dynamics of variable-mass systems and show yet another perspective of the subject.