Momentum

Momentum is a physical quantity associated with motion and a fundamental conserved quantity in physics. In classical mechanics, a particle’s linear momentum is the product of its mass and velocity, $\mathbf p=m\mathbf v$. It has both magnitude and direction. For an isolated system, total momentum remains constant even when its parts interact. Relativistic and quantum theories retain momentum’s central role while extending its definition and mathematical representation. (charma.uprm.edu)

Definition and measurement

For a classical particle, momentum points in the direction of its velocity, and its magnitude increases proportionally with both mass and speed. Its SI unit is kilogram metre per second, $\mathrm{kg\,m\,s^{-1}}$, equivalent to a newton second. Momentum depends on the reference frame: an object at rest relative to one observer can have nonzero momentum relative to another. The momenta of several objects must therefore be expressed in the same frame before they are added. (charma.uprm.edu)

Momentum is distinct from kinetic energy. For a nonrelativistic particle of positive mass,

Kinetic energy is a scalar, whereas momentum is a vector. Two equal masses moving at equal speeds in opposite directions have zero combined momentum but positive combined kinetic energy. Consequently, cancellation of momenta does not imply an absence of motion or energy. (charma.uprm.edu)

Force and impulse

Newton’s second law relates the net external force acting on a particle to the rate of change of its momentum:

For constant mass, this becomes $\mathbf F_{\mathrm{net}}=m\mathbf a$. A force can change momentum by changing speed, direction, or both; motion around a circle, for example, involves continuously changing momentum even at constant speed. (openstax.org)

The impulse delivered over a time interval is the time integral of the net force:

This impulse–momentum theorem allows collisions to be analyzed without knowing every detail of the rapidly varying contact force. For a given momentum change, increasing the interaction time reduces the magnitude of the average net force. This relationship underlies the operation of impact-absorbing materials and vehicle restraints. (openstax.org)

Conservation and collisions

For a system of particles, total momentum is the vector sum

In Newtonian mechanics, equal and opposite internal forces cancel when the particles’ momentum changes are added. Hence $d\mathbf P/dt=\mathbf F_{\mathrm{ext}}$. If the net external force vanishes, total momentum is conserved; if external impulse is negligible during a brief interaction, conservation is a useful approximation. Conservation applies separately to each spatial component, not to the sum of momentum magnitudes. (openstax.org)

In an elastic collision, total kinetic energy and total momentum are conserved. In an inelastic collision, kinetic energy is not conserved, although total energy remains conserved when deformation, heating, and other forms are included. Objects stick together in a perfectly inelastic collision. For two classical bodies with negligible external impulse, their common final velocity is

Recoil and explosions likewise redistribute momentum among a system’s parts without changing its total. A gun and bullet, for example, acquire oppositely directed momenta when external impulse is negligible, while stored chemical energy supplies their kinetic energy. (openstax.org)

Symmetry and canonical momentum

A deeper explanation of momentum conservation comes from Noether’s theorem: continuous spatial translation symmetry produces a conserved momentum. When shifting an entire isolated system does not change its governing laws, there is a corresponding conservation law. Momentum also acts as the generator of spatial translations in quantum theory. (ocw.mit.edu)

In analytical mechanics, the momentum conjugate to a generalized coordinate $q_i$ is defined using the Lagrangian:

This canonical momentum need not equal mechanical momentum. For a nonrelativistic particle with electric charge $Q$ in an electromagnetic field, the usual SI expression is

where $\mathbf A$ is the magnetic vector potential. Mechanical momentum remains $m\mathbf v$. The distinction is important when constructing the Hamiltonian for charged particles and describing their interaction with electromagnetic fields. (ocw.mit.edu)

Relativistic momentum

In special relativity, a particle with rest mass $m$ has momentum

where $c$ is the speed of light in vacuum. At speeds much smaller than $c$, this approaches the classical expression. For a massive particle, momentum grows without bound as its speed approaches $c$. (openstax.org)

Total energy and momentum obey

They form the components of four-momentum, which transforms between inertial frames under a Lorentz transformation. A photon has zero rest mass but nonzero momentum: $p=E/c$. Thus the classical formula $m\mathbf v$ is not a universal definition, and massless radiation can transfer momentum to matter. (openstax.org)

Quantum momentum

In quantum mechanics, canonical momentum is represented in the position representation by the operator

acting on the wave function, with $\hbar=h/(2\pi)$ and $h$ the Planck constant. A general quantum state need not possess a definite momentum; instead, momentum measurements have a probability distribution determined by that state. (ocw.mit.edu)

The de Broglie relation connects momentum magnitude to wavelength, $\lambda=h/p$. Position and the corresponding momentum component satisfy the uncertainty principle,

where the uncertainties denote standard deviations. This constrains their simultaneous statistical sharpness rather than merely describing imperfect measuring instruments. (openstax.org)