Using modern symbolic notation, Newton's second law can be written as a vector differential equation:
where:
is the net force vector
is mass
is the velocity vector
is time.
The product of the mass and velocity is the momentum of the object (which Newton himself called "quantity of motion"). The use of algebraic expressions became popular the 18th Century, after Newton's death, while vector notation dates to the late 19th Century. The Principia expresses mathematical theorems in words and consistently uses geometrical rather than algebraic proofs.
If the mass of the object in question is constant, this differential equation can be rewritten as:
where:
A verbal equivalent of this is "the acceleration of an object is proportional to the force applied, and inversely proportional to the mass of the object". If m is dependent on velocity (and thus indirectly upon time) as we now know it is (for high velocities—see special relativity), then m has to be included in the derivative, as above.
Taking Special Relativity into consideration, the equation will become
where:
m0 is the rest mass or invariant mass.
Note that force will depend on speed of the moving body, acceleration and its rest mass. However, when the speed of the moving body is much lower than the speed of light, the equation that was shown above will be reduced to our familiar
Contrary to what is sometime claimed in elementary texts, mass must always be taken as constant in classical mechanics. So-called variable mass systems like a rocket can not be directly treated by making mass a function of time in the second law. The reasoning, given in An Introduction to Mechanics by Kleppner and Kolenkow and other modern texts, is excerpted here:
Newton's second law applies fundamentally to particles. In classical mechanics, particles by definition have constant mass. In case of well-defined systems of particles, Newton's law can be extended by integrating over all the particles in the system. In this case we have to refer all vectors to he center of mass. Applying the second law to extended objects implicitly assumes the object to be a well-defined collection of particles. However, 'variable mass' systems like a rocket or a leaking bucket do not consist of a set number of particles. They are not well-defined systems. Therefore Newton's second law can not be applied to them directly. The naive application of F = dp/dt will usually result in wrong answers in such cases. However, applying the conservation of momentum to a complete system (such as rocket+fuel, or bucket+leaked water) will give unambiguously correct answers.
