In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object. For example, a heavy truck moving fast has a large momentum—it takes a large and prolonged force to get the truck up to this speed, and it takes a large and prolonged force to bring it to a stop afterwards.

- Like velocity, linear momentum is a vector quantity, possessing a direction as well as a magnitude: Linear momentum is also a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum cannot change.
- Although originally expressed in Newton’s third law, the conservation of linear momentum also holds in special relativity and, with appropriate definitions, a (generalized) linear momentum conservation law holds in electrodynamics, quantum mechanics, quantum field theory, and general relativity.
- Because momentum has a direction, it can be used to predict the resulting direction of objects after they collide, as well as their speeds.
- It is the product of two quantities, the mass (represented by the letter m) and velocity (v): The units of momentum are the product of the units of mass and velocity.
- If two particles have masses m1 and m2, and velocities v1 and v2, the total momentum is The momenta of more than two particles can be added in the same way.
- If a force F is applied to a particle for a time interval Δ t, the momentum of the particle changes by an amount In differential form, this gives Newton’s second law: the rate of change of the momentum of a particle is equal to the force F acting on it: If the force depends on time, the change in momentum (or impulse) between times t1 and t2 is If the mass is constant, so the force is equal to mass times acceleration.
- The thrust required to produce this acceleration is 1 newton.
- This fact, known as the law of conservation of momentum, is implied by Newton’s laws of motion.
- This independence of reference frame is called Newtonian relativity or Galilean invariance.
- By itself, the law of conservation of momentum is not enough to determine the motion of particles after a collision.
- If it is conserved, the collision is called an elastic collision; if not, it is an inelastic collision.
- Elastic collisions An elastic collision is one in which no kinetic energy is lost.
- Now if a given coordinate qi does not appear in the Lagrangian, then This is the generalization of the conservation of momentum.
- An example is found in the section on electromagnetism.
- The Galilean transformation gives the coordinates of the moving frame as while the Lorentz transformation gives where γ is the Lorentz factor: Newton’s second law, with mass fixed, is not invariant under a Lorentz transformation.
- If Pmech is the momentum of all the particles in a volume V, and the particles are treated as a continuum, then Newton’s second law gives The electromagnetic momentum is and the equation for conservation of energy for each component i of the momentum is The term on the right is an integral over the surface S representing momentum flow into and out of the volume, and nj is a component of the surface normal of S. The quantity Ti j is called the Maxwell stress tensor, defined as The above results are for the microscopic Maxwell equations, applicable to electromagnetic forces in a vacuum.
- In quantum mechanics, momentum is defined as an operator on the wave function.