Can momentum be hidden to human eyes like how kinetic energy can be hidden as heat?
Category: Physics Published: March 19, 2014
Yes and no. In a regular mechanical system with macroscopic parts, momentum can not be "hidden" to human eyes. But in other systems, momentum can be hidden. For instance, in an electromagnetic system, momentum can be transferred to the electromagnetic field, which is invisible to human eyes at most frequencies. Therefore, momentum can be "hidden" in the electromagnetic field. Let us explore this topic more in depth.
Every isolated system obeys the law of conservation of energy which states that if no energy is externally extracted or inserted into the system, its total energy will remain constant in time. For example, consider two hockey pucks sliding towards each other on ice. If you sum the kinetic (motional) energy of both pucks before they collide, you will find that it equals the sum of both energies after they collide. We can in fact use the conservation of energy along with some other information to predict what will happen in a simple collision like this.
But now suppose we fire the two pucks directly at each other at the same speed and cover their sides with perfect glue. What happens? Of course, when the pucks make contact, they stick together and both end up with zero speed, and therefore zero kinetic energy. This means that the total kinetic energy of the two-puck system at the beginning was some big number, but the total kinetic energy at the end was zero (at least according to human eyes). Has the glue broken the law of conservation of energy? No. If you did a careful analysis of this event, and measured everything you could think of, you would find that the heat (thermal energy) in the two pucks increases after the collision by the exact same amount as the kinetic energy that seemed to disappear (neglecting the friction of the ice). The macroscopic kinetic energy has therefore been converted into heat. The law of conservation of energy still holds as long as we add heat as one of the things that contributes to the total energy. Such a collision is called an inelastic collision.
But what is heat? On the atomic level, we find that hotter substances have their atoms vibrating faster and moving around at higher speeds. Thermal energy is therefore just the kinetic energy of microscopic, randomly-moving particles. When the two pucks with glue stick together, their kinetic energy is not really converted to some mysterious thing called "heat". Their macroscopic, ordered kinetic energy (ordered in the sense that all the atoms in the puck are moving along with the puck as it zips across the ice) is simply converted to microscopic, random kinetic energy. The atoms at the surface of each puck smash together, get displaced, smash into other atoms, and so on, creating vibrations. Macroscopic, ordered kinetic energy is obvious to the human eye (we see the puck moving), but microscopic, random kinetic energy is invisible to the human eye (we cannot directly see the atoms jiggling). In this sense, an inelastic collision causes some of the kinetic energy to be "hidden" from human eyes in the form of heat. If we include only the forms of energy visible to the human eye, glue seems to defeat the law of conservation of energy. But if we include hidden kinetic energy (heat), the law still holds.
Now there is another law called the law of conservation of momentum. It states that the total momentum of an isolated system before an event must be equal to the total momentum of the system after the event. What is the difference between this law and the law of conservation of energy. The difference is that energy is just a number, while momentum has a direction. Conservation of momentum therefore tells us things like: two pucks initially traveling to the right that collide (say, because one is going faster and overtakes the other) must still be traveling to the right after the collision. The total momentum of both pucks is to the right in the beginning, so it must be to the right in the end. The law of conservation of energy cannot tell us this because it says nothing about directions. Or similarly, if one puck traveling east approaches another puck traveling north, the total momentum is in the north-east direction. Conservation of momentum tells us that after the collision, the total momentum must still be in the north-east direction.
Since momentum obeys a conservation law just like energy, and since kinetic energy can be hidden from human eyes in the form of microscopic motion, it is natural to wonder whether momentum can also be hidden. Does the two-puck-glue system convert momentum to some hidden form upon collision, and therefore some momentum seems to get "lost" to human eyes? No. The reason the answer is no is because momentum is directional. The atoms do indeed jiggle faster after an inelastic collision, but the total momentum of atoms jiggling in place is always zero. Put simply, when the atomic motion is random, for every atom going left there is another atom going right, so that their directionality adds to zero. Since momentum describes the directionality of motion, the momentum for simple thermal motion is zero. In order for momentum to not be zero, the atoms have to all be traveling more or less in the same direction. But when all the atoms are traveling in the same direction, the macroscopic object itself is traveling somewhere, which is quite visible to the human eye. Because momentum is directional, it cannot be hidden from the human eye in the form of random atomic motion. Therefore, even if we cover the sides of the pucks with glue so that they stick, the total visible momentum before will equal the total visible momentum after. This makes sense because if the two pucks are flying at each other at the same speed from opposite directions (and have the same mass), their total momentum before is zero (east plus the same amount of west equals zero), so their total momentum after sticking together is zero (zero plus zero equals zero).
This idea extends beyond pucks colliding. Any macroscopic mechanical system cannot hide momentum. But, if the system has parts that are so small that they are invisible to the naked eye, and these parts can undergo ordered motion, momentum can be transferred and hidden in these parts. For instance, if a tennis ball is covered with dust grains that are so small you can't see them, when the tennis ball gets hit, the dust can fly off to one side and carry away some of the momentum. If we don't notice the dust, and don't include it in our calculations, we would find that some of the total momentum becomes hidden after the collision.
Richard Feynman states in his book, The Feynman Lectures on Physics, the following:
"Are there also hidden forms of momentum-perhaps heat momentum?" The answer is that it is very hard to hide momentum for the following reasons. The random motions of the atoms of a body furnish a measure of heat energy, if the squares of the velocities are summed. This sum will be a positive result, having no directional character. The heat is there, whether or not the body moves as a whole, and conservation of energy in the form of heat is not very obvious. On the other hand, if one sums the velocities, which have direction, and finds a result that is not zero, that means that there is a drift of the entire body in some particular direction, and such a gross momentum is readily observed. Thus there is no random internal lost momentum, because the body has net momentum only when it moves as a whole. Therefore momentum, as a mechanical quantity, is difficult to hide. Nevertheless, momentum can be hidden-in the electromagnetic field, for example.
Let us move beyond mechanical systems. Electromagnetic waves such as light or radio waves carry momentum. Since all electromagnetic waves are invisible to humans except the narrow range of colors from red to violet, momentum can be hidden in the electromagnetic field. In practice, the momentum carried by electromagnetic waves is so small that you have to use very sensitive instruments to measure it, but in principle it is always there. For instance, if you turn on a radar gun, radio waves come out the front of the gun, carrying momentum with them. Because of momentum conservation, the radar gun itself must therefore recoil in the opposite direction when it is turned on. This recoil is typically too small to notice. But for the sake of the argument, suppose we have a giant radar gun that emits a large amount of radio waves. Since radio waves are invisible to the human eye, all that we would see would be the gun jump backwards upon being turned on, clearly violating conservation of momentum (if we only include visible momentum). We would therefore conclude (and rightly so), that there must be momentum hidden somewhere. In this way, momentum can be "hidden" in the electromagnetic field. Although, it is really only hidden to human eyes. We can detect radio waves just fine with an antenna.