## Key Concepts

**The fundamental physical law stating that the momentum of a system is constant if no external forces act upon the system.** The principle of conservation of momentum holds generally and is applicable in all fields of physics. In particular, momentum is conserved even if the particles of a system exert forces on one another or if the total mechanical energy is not conserved. Use of the principle of conservation of momentum is fundamental in the solution of collision problems (Fig. 1).* See also: ***Collision (physics)**; **Conservation of energy**; **Conservation laws (physics)**; **Energy**; **Momentum**; **Physics**

### Conservation of momentum examples

#### Standing on a cart

If a person standing on a well-lubricated cart steps forward, the cart moves backward. One can explain this result by momentum conservation, considering the system to consist of cart and human. If both person and cart are originally at rest, the momentum of the system is zero. If the person then acquires forward momentum by stepping forward, the cart must receive a backward momentum of equal magnitude in order for the system to retain a total momentum of zero.

#### An astronaut and a satellite

When the principle of conservation of momentum is applied, care must be taken that the system under consideration is in fact isolated from external forces. For example, when a rough rock rolls down a hill, the isolated system would have to consist of the rock plus the Earth, and not the rock alone, since momentum exchanges between the rock and the Earth cannot be neglected. In a more ideal conception of an isolated system, consider an astronaut in space touching a satellite. When the astronaut extends his or her hands, pushing against and away from the satellite, the satellite likewise moves away from the point of contact, with an equal magnitude to the force exerted by the astronaut's muscles. * See also: ***Earth**; **Satellite (spacecraft)**; **Space**

##### Video example

#### Rocket propulsion

The propulsion of a rocket through space can also be explained in terms of momentum conservation. Hot gases produced by the combustion of the fuel are expelled at high speed from the rear of the rocket. Although the total mass of these hot gases may not be large, the gases move with such a high velocity that the total momentum associated with them is appreciable. The momentum of the gases is directed backward. For momentum to be conserved, the rocket must acquire an equal momentum in the forward direction. If the rocket carries all the materials needed for the combustion of its fuel, its propulsion does not require air, and it can move through empty space. * See also: ***Air**; **Gas**; **Mass**; **Propulsion**; **Rocket**

#### Exploding bomb

An exploding bomb gives another application of the conservation of momentum. The total resultant vector momentum of all the pieces of the bomb immediately after explosion must equal the momentum of the unexploded bomb just before the explosion. While the previous examples were one-dimensional, the exploding bomb generally involves conservation of momentum in more than one spatial dimension simultaneously. In practice, solving this example mathematically typically involves breaking down the momentum of the bomb itself (before exploding) and each piece of the bomb (after exploding) into components for each axis of motion (x, y, and z axes), then applying momentum conservation for each axis. For instance, if solving this problem with exploding bomb fragments moving in two dimensions, one would use trigonometric functions to determine the components of momentum for each object in the x- and y- directions, then write out and solve equations for conservation of momentum in the x-direction and conservation of momentum in the y-direction. * See also: ***Explosive**

### Proof of principle

The principle of conservation of momentum follows directly from Newton's second and third laws. While the principle will be proved here only for the straight-line motion of a two-particle system, it can be generalized to systems containing any number of particles. A particle is a mass with dimensions so small that rotational effects are negligible. Momentum will also be conserved for rigid bodies large enough that rotation must be considered, since rigid bodies can be treated as assemblies of many particles. * See also: ***Newton's laws of motion**

For the one-dimensional motion of an isolated two-particle system, Newton's third law states that the force *F*_{12} that particle 1 exerts on particle 2 is equal in magnitude and opposite in direction to the force *F*_{21} that particle 2 exerts on particle 1. Thus Eq. (1) holds.

By use of Newton's second law this equation can be expressed in terms of the momenta *m*_{1}
*v*_{1} and *m*_{2}
*v*_{2} of particles 1 and 2, respectively, where *m*_{1}, *m*_{2}, *v*_{1}, and *v*_{2} are the masses and velocities of particles 1 and 2, respectively. Then Eq. (2) holds.

Integration gives Eq. (3),

where *c* is a constant. This equation expresses the conservation of momentum for two particles moving in the same straight line.

Angular and linear momentum are independent quantities. A complete description of a system must include both quantities. The angular momentum of a system is conserved under quite general conditions.* See also: ***Angular momentum**