# Momentum

Momentum is a property of motion that in classical physics is a vector (directional) quantity that in closed systems is conserved during collisions. In Newtonian physics momentum is measured as the product of the mass and component velocity of a body. For massless particles (e.g., photons) moving at the speed of light (v = c) the momentum (p) is equal to Planck's constant divided by the wavelength.

The first formal definitions and measurement of momentum date to the writing of French philosopher René Descartes (1596–1650). Descartes intended momentum to a quantifiable and measurable concept related to what he termed the "amount of motion."

Measurement of momentum often concentrates on rates of change in the momentum of bodies. In accord with the law of inertia, a body with no net force acting upon it experiences no change in momentum and therefore measurement of momentum reflect that momentum is conserved. Whenever a net force is applied to a body the change in momentum is proportional to the force applied but the conservation of momentum dictates that the momentum of the agent applying force to the body must correspondingly decrease so that the measured momentum of the combined systems remains unchanged.

Modern devices used to measure momentum of subatomic particles often employ tracking devices located in strong magnetic fields. The paths of particles moving through these fields reveals their charge and momentum. The direction of deflection reveals the particles change and the momenta of particles can be calculated from the fact that the paths of particles with greater momentum deviate less than those of lesser momentum (i.e., those particles with higher momentum tend to travel along straighter or less bent paths).

Quantum theory dictates that the measurement of certain pairs of properties of particles, including position and momentum are limited by the Heisenberg uncertainty principle first advanced by German physicist Werner Heisenberg (1901-1976). In essence, although it is possible to measure either position or momentum the pair can not be measured simultaneously. The more exact the determination of position, the more uncertain becomes the measurement of momentum.

Although the uncertainty principle is not relevant to the measurement of momentum of large objects, it places severe constraints on measurements of momentum of subatomic particles. Accordingly, quantum theory places a limitation on the experimental measurement of momentum. The more accuracy required in the determination of position, the less the accuracy possible with regard to the determination of momentum. For example, in attempting to make an accurate determination of the position of an electron it is necessary to bombard the electron with photons. In doing so the collisions between the photons and the electron alter the momentum of the electron and therefore introduce uncertainty in the measurement of the momentum of the electron.

Moreover, there are important philosophical ramifications to the measurement of momentum, In the Copenhagen interpretation of quantum mechanics, reality is dependent upon the observer's measurement. Essentially, the Copenhagen interpretation dictates that in the measurement of momentum or position of a two particle system, the measurement of momentum or position of one particle gives reality to the momentum or position of the second particle. In this theoretical interpretation conflicting realities result when there is an attempt to measure the momentum of one particle and the position of the other. Because time-ordering of the measurements is dependent upon the inertial frame varying reference frames yield differing realities and give rise to a problem in nonlocality related to the instantaneous propagation of information related to the measurement across and real space.

The momentum of an object is the mass of the object multiplied by the velocity of the object. The mass will often be measured in kilograms (kg) and the velocity, in meters per second (m/s), so the momentum will be measured in kilogram meters per second (kg m/s). Because velocity is a vector quantity, meaning that the direction is part of the quantity, momentum is also a vector. Just like the velocity, to completely specify the momentum of an object one must also give the direction.

A force multiplied by the length of time that the force acts is called the impulse. According to the impulse momentum theorem, the impulse acting on an object is equal to the change in the object's momentum. Notice the word change. The impulse is not equal to the object's momentum, but the amount the momentum changes. (This impulse momentum theorem is basically a disguised form of Newton's second law.) The force used to figure out the impulse here is the total sum of all the external forces acting on an object. Internal forces acting within an object do not count.

The consequences of this impulse momentum theorem are rather profound. If there are no external forces acting on an object, then the impulse (force times time) is zero. The change in momentum is also zero because it is equal to the force. Hence, if an object has no external forces acting on it, the momentum of the object can never change. This law is the law of conservation of momentum. There are no known exceptions to this fundamental law of physics. Like other conservation laws (such as conservation of energy), the law of conservation of momentum is a very powerful tool for understanding the universe.

## Resources

### Books

De Gosson, Maurice. Principles of Newtonian and Quantum Mechanics. River Edge, NJ: World Scientific Publishers, 2001.

Moore, Thomas. Six Ideas that Shaped Physics: Unit C; Conservation Laws Constrain Actions. New York: McGraw-Hill, 2002.

### Other

The Physics Classroom and Mathsoft Engineering & Education, Inc. "The Law of Momentum Conservation" [cited March 10, 2003]. <http://www.physicsclassroom.com/Class/momentum/U4L2a.html>.