Momentum

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MOMENTUM

Momentum and energy are among the most important quantities in physics. Their importance arises from the fact that they are conserved, which means that energy is never created or destroyed, although it can be transformed from one form to another. There is always an exact accounting so that in the end the books balance to exactly zero. For example, the Earth absorbs solar energy, but the energy is transformed into thermal energy of the Earth. Most of this energy is radiated back into space, but, if the Earth is warming, some remains as thermal energy. When all such energy changes are added up, the result is always zero net energy change. This is conservation of energy. Momentum obeys a similar conservation law.

Definition of Momentum

While energy is a scalar (a number), momentum is a vector. A moving particle has a momentum,p , equal to its mass, m , times its velocity, v —p = mv . The velocity vector v can be represented by an arrow. The length of the arrow equals the speed of the particle—how fast it is moving and the direction of the arrow indicates the direction of the motion. When v is multiplied by the mass m , the result is the momentum vector, which points in the same direction as the velocity vector. Momentum is defined this way because, with this definition, momentum is conserved.

The Law of Conservation of Momentum

Conservation of momentum is illustrated in Figure 1. A moving particle of mass, m , strikes a stationary particle of mass, M . Initially, the total momentum of the system is simply the momentum of the moving particle, mv . After the collision, the initial momentum is shared between the two particles as shown. Although they may be moving at wildly different speeds and directions, by the law of momentum conservation the sum of the two final momenta equals that of the original incoming particle. To add the two vectors, one places the vectors head to tail (Figure 1b): the sum is the vector drawn from the tail of the first to the head of the second. The figure shows that the sum of the final momenta equals the original momentum.

FIGURE 1

According to relativity any ordinary vector is always paired with a scalar quantity to form a four-vector. The most familiar case is the space-time four-vector consisting of position (described by a vector) and time. Momentum (a vector) and energy (a scalar) also form a four-vector. Relativity requires that all components of a four-vector be conserved if any of them are. Thus, conservation of energy and momentum are really the same conservation law, the conservation of the energy-momentum four-vector.

Symmetries and Conservation Laws

All conservation laws are believed to come from symmetries (Noether's Theorem). Energy conservation comes from the symmetry that the laws of nature are the same at all times. This is a symmetry in the same sense that a circle is symmetric because it is the same no matter how you rotate it. Changing the angle of rotation changes nothing; for the time symmetry we change time and nothing changes, that is, the laws of physics stay the same.

Relativity suggests that position (the three-vector part of the space-time four-vector) should show a similar symmetry and that this symmetry should give rise to the conservation of momentum. Indeed, this is the case: the laws of physics are unchanged when we move from one place to another. Otherwise, physicists in Hong Kong would have to use different laws and theories than physicists in Canada. But they do not, and conservation of momentum results.

Colliding beam accelerators

Particle accelerators are designed to produce new particles in order to test predictions of theories of elementary particles. For example, the Higgs boson is predicted by current models, and its discovery would be a major confirmation of those theories. The primary reason for wanting to build the Superconducting Super Collider, which was canceled in 1993, was to look for the Higgs boson. Europe's Large Hadron Collider is designed to carry out the same search.

The demands of conservation of momentum are a significant obstacle for particle production and have required a major redesign of particle accelerators. New particles are produced by accelerating familiar particles to large energies and aiming the beam at a stationary target. When a beam particle strikes a target particle, new particles can be formed if there is enough energy available to create the new particle, that is, to create its rest energy. Ideally, we would like all of the kinetic energy of the incoming particle plus the rest energies of the two initial particles to go into creating the new particle.

The problem is that momentum conservation requires that there be some net forward momentum of the new particle equal to the momentum of the incoming particle (Figure 2). Thus, some of the energy must go into this motion and is not available for creating the new particle. For new particles with large masses like the Higgs boson (about 1,000 times the mass of a proton), only a tiny fraction of the incoming particle's energy is available for particle creation; the vast majority of the energy is used up in satisfying conservation of momentum. For example, in producing a particle with a mass of 50 times that of a proton, only 4 percent of the incoming particle's energy is available, meaning that the beam particle must have a kinetic energy of 1,250 times the rest energy of a proton, or about 1,200 GeV. This exceeds the capabilities of the most energetic accelerators, and creating the heavier Higgs boson is even further out of reach.

The solution is straightforward, in principle. Instead of a beam of particles colliding with a stationary target, let two beams of identical particles collide head-on (Figure 3). The net initial momentum is zero so the final momentum is also zero. No energy

FIGURE 2

FIGURE 3

has to go into post-collision kinetic energy so all of the kinetic energy of both beam particles plus their rest energies are available to create the new particle.

Still, there is a downside. Since the spacing between particles in a beam is much greater than in a material target, the rate of collisions is correspondingly less. Thus, one has to run the experiment for a long time in order to produce and detect the desired particle. Nonetheless, this is far better than not being able to produce the particles at all.