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It is the product of two quantities, the particle's mass (represented by the letter m) and its velocity (v): [1] =. The unit of momentum is the product of the units of mass and velocity. In SI units, if the mass is in kilograms and the velocity is in meters per second then the momentum is in kilogram meters per second (kg⋅m/s).
The Lagrangian indicates an additional detail: the canonical momentum in Lagrangian mechanics is given by: = ˙ = ˙ + instead of just mv, implying the motion of a charged particle is fundamentally determined by the mass and charge of the particle. The Lagrangian expression was first used to derive the force equation.
Internal forces between the particles that make up a body do not contribute to changing the momentum of the body as there is an equal and opposite force resulting in no net effect. [3] The linear momentum of a rigid body is the product of the mass of the body and the velocity of its center of mass v cm. [1] [4] [5]
The concepts invoked in Newton's laws of motion — mass, velocity, momentum, force — have predecessors in earlier work, and the content of Newtonian physics was further developed after Newton's time. Newton combined knowledge of celestial motions with the study of events on Earth and showed that one theory of mechanics could encompass both.
The motion of a particle under a central force F always remains in the plane defined by its initial position and velocity. [7] This may be seen by symmetry. Since the position r , velocity v and force F all lie in the same plane, there is never an acceleration perpendicular to that plane, because that would break the symmetry between "above ...
The equation of motion for a particle of constant mass m is Newton's second law of 1687, in modern vector notation =, where a is its acceleration and F the resultant force acting on it. Where the mass is varying, the equation needs to be generalised to take the time derivative of the momentum.
A free particle with mass in non-relativistic quantum mechanics is described by the free Schrödinger equation: (,) = (,) where ψ is the wavefunction of the particle at position r and time t . The solution for a particle with momentum p or wave vector k , at angular frequency ω or energy E , is given by a complex plane wave :
In relativistic mechanics, the relativistic angular momentum of a particle is expressed as an anti-symmetric tensor of second order: = in terms of four-vectors, namely the four-position X and the four-momentum P, and absorbs the above L together with the moment of mass, i.e., the product of the relativistic mass of the particle and its center ...