0.3 6.4 conservation of momentum  (Page 2/5)

where ${p\prime }_1$ and ${p\prime }_2$ are the momenta of cars 1 and 2 after the collision. (We often use primes to denote the final state.)

This result—that momentum is conserved—has validity far beyond the preceding one-dimensional case. It can be similarly shown that total momentum is conserved for any isolated system, with any number of objects in it. In equation form, the conservation of momentum principle    for an isolated system is written

or

where ${\mathbf{p}}_{\text{tot}}$ is the total momentum (the sum of the momenta of the individual objects in the system) and ${\mathbf{\text{p}}\prime }_{\text{tot}}$ is the total momentum some time later. (The total momentum can be shown to be the momentum of the center of mass of the system.) An isolated system    is defined to be one for which the net external force is zero $\left({\mathbf{\text{F}}}_{\text{net}}=0\right)\text{.}$

Perhaps an easier way to see that momentum is conserved for an isolated system is to consider Newton’s second law in terms of momentum, ${\mathbf{F}}_{\text{net}}=\frac{{\mathrm{\Delta }\mathbf{p}}_{\text{tot}}}{\mathrm{\Delta }t}$ . For an isolated system, $\left({\mathbf{\text{F}}}_{\text{net}}=0\right)$ ; thus, $\mathrm{\Delta }{\mathbf{p}}_{\text{tot}}=0$ , and ${\mathbf{p}}_{\text{tot}}$ is constant.

We have noted that the three length dimensions in nature— $x$ , $y$ , and $z$ —are independent, and it is interesting to note that momentum can be conserved in different ways along each dimension. For example, during projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero and momentum is unchanged. But along the vertical direction, the net vertical force is not zero and the momentum of the projectile is not conserved. (See [link] .) However, if the momentum of the projectile-Earth system is considered in the vertical direction, we find that the total momentum is conserved.

The conservation of momentum principle can be applied to systems as different as a comet striking Earth and a gas containing huge numbers of atoms and molecules. Conservation of momentum is violated only when the net external force is not zero. But another larger system can always be considered in which momentum is conserved by simply including the source of the external force. For example, in the collision of two cars considered above, the two-car system conserves momentum while each one-car system does not.

Subatomic collisions and momentum

The conservation of momentum principle not only applies to the macroscopic objects, it is also essential to our explorations of atomic and subatomic particles. Giant machines hurl subatomic particles at one another, and researchers evaluate the results by assuming conservation of momentum (among other things).

On the small scale, we find that particles and their properties are invisible to the naked eye but can be measured with our instruments, and models of these subatomic particles can be constructed to describe the results. Momentum is found to be a property of all subatomic particles including massless particles such as photons that compose light. Momentum being a property of particles hints that momentum may have an identity beyond the description of an object’s mass multiplied by the object’s velocity. Indeed, momentum relates to wave properties and plays a fundamental role in what measurements are taken and how we take these measurements. Furthermore, we find that the conservation of momentum principle is valid when considering systems of particles. We use this principle to analyze the masses and other properties of previously undetected particles, such as the nucleus of an atom and the existence of quarks that make up particles of nuclei. [link] below illustrates how a particle scattering backward from another implies that its target is massive and dense. Experiments seeking evidence that quarks make up protons (one type of particle that makes up nuclei) scattered high-energy electrons off of protons (nuclei of hydrogen atoms). Electrons occasionally scattered straight backward in a manner that implied a very small and very dense particle makes up the proton—this observation is considered nearly direct evidence of quarks. The analysis was based partly on the same conservation of momentum principle that works so well on the large scale.

Section summary

• The conservation of momentum principle is written
  ${\mathbf{p}}_{\text{tot}}=\text{constant}$
  or
  ${\mathbf{\text{p}}}_{\text{tot}}={\mathbf{\text{p}}\prime }_{\text{tot}}(\text{isolated system}),$
  ${\mathbf{p}}_{\text{tot}}$ is the initial total momentum and ${\mathbf{\text{p}}\prime }_{\text{tot}}$ is the total momentum some time later.
• An isolated system is defined to be one for which the net external force is zero $\left({\mathbf{\text{F}}}_{\text{net}}=0\right)\text{.}$
• During projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero.
• Conservation of momentum applies only when the net external force is zero.
• The conservation of momentum principle is valid when considering systems of particles.

Conceptual questions

Problems&Exercises