Variable-mass systems

Variable-mass systems, like a rocket burning fuel and ejecting spent gases, are not closed and cannot be directly treated by making mass a function of time in the second law.^[17] The reasoning, given in An Introduction to Mechanics by Kleppner and Kolenkow and other modern texts, is that Newton's second law applies fundamentally to particles.^[18] In classical mechanics, particles by definition have constant mass. In case of a well-defined system of particles, Newton's law can be extended by summing over all the particles in the system:

where F_net is the total external force on the system, M is the total mass of the system, and a_cm is the acceleration of the center of mass of the system.

Variable-mass systems like a rocket or a leaking bucket cannot usually be treated as a system of particles, and thus Newton's second law cannot be applied directly. Instead, the general equation of motion for a body whose mass m varies with time by either ejecting or accreting mass is obtained by rearranging the second law and adding a term to account for the momentum carried by mass entering or leaving the system:^[16]

where u is the relative velocity of the escaping or incoming mass with respect to the center of mass of the body. Under some conventions, the quantity u dm/dt on the left-hand side, known as the thrust, is defined as a force (the force exerted on the body by the changing mass, such as rocket exhaust) and is included in the quantity F. Then, by substituting the definition of acceleration, the equation becomes

History of the second law

Newton's Latin wording for the second law is:

Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.

This was translated quite closely in Motte's 1729 translation as:

LAW II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.

According to modern ideas of how Newton was using his terminology,^[23] this is understood, in modern terms, as an equivalent of:

The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed.

Motte's 1729 translation of Newton's Latin continued with Newton's commentary on the second law of motion, reading:

If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

The sense or senses in which Newton used his terminology, and how he understood the second law and intended it to be understood, have been extensively discussed by historians of science, along with the relations between Newton's formulation and modern formulations