|Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....|
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Although velocity has been referred to, it is not the total velocity of a closed system that remains constant. If it was, then firing a gun would cause the gun to recoil at the same velocity as the bullet! The gun does recoil, but at a much lower velocity than the bullet. Newton's third law tells us
and assuming a constant force for simplicity, Newton's second law allows us to change this to
Thus if the gun has 100 times more mass than the bullet, it will recoil at a velocity that is 100 times smaller and in the opposite direction, represented by the opposite sign. The quantity mv is therefore apparently a useful measure of motion, and we give it a name, momentum, and a symbol, p. (As far as I know, the letter p was just chosen at random, since m was already being used for mass.) The situations discussed so far have been one-dimensional, but in three-dimensional situations it is treated as a vector.
definition of momentum for material objects
The momentum of a material object, i.e., a piece of matter, is defined as
p = mv ,
the product of the object's mass and its velocity vector.
The units of momentum are kg·m/s, and there is unfortunately no abbreviation for this clumsy combination of units.
The reasoning leading up to the definition of momentum was all based on the search for a conservation law, and the only reason why we bother to define such a quantity is that experiments show it is conserved:
the law of conservation of momentum
In any closed system, the vector sum of all the momenta remains constant,
p1i + p2i + . . . = p1f + p2f + . . . ,
where i labels the initial and f the final momenta. (A closed system is one on which no external forces act.)
This chapter first addresses the one-dimensional case, in which the direction of the momentum can be taken into account by using plus and minus signs. We then pass to three dimensions, necessitating the use of vector addition.
A subtle point about conservation laws is that they all refer to closed systems, but closed means different things in different cases. When discussing conservation of mass, closed means a system that doesn't have matter moving in or out of it. With energy, we mean that there is no work or heat transfer occurring across the boundary of the system. For momentum conservation, closed means there are no external forces reaching into the system.
Generalization of the momentum concept
As with all the conservation laws, the law of conservation of momentum has evolved over time. In the 1800's it was found that a beam of light striking an object would give it some momentum, even though light has no mass, and would therefore have no momentum according to the above definition. Rather than discarding the principle of conservation of momentum, the physicists of the time decided to see if the definition of momentum could be extended to include momentum carried by light. The process is analogous to the process outlined on page 19 for identifying new forms of energy. The first step was the discovery that light could impart momentum to matter, and the second step was to show that the momentum possessed by light could be related in a definite way to observable properties of the light. They found that conservation of momentum could be successfully generalized by attributing to a beam of light a momentum vector in the direction of the light's motion and having a magnitude proportional to the amount of energy the light possessed. The momentum of light is negligible under ordinary circumstances, e.g., a flashlight left on for an hour would only absorb about 10-5 kg·m/s of momentum as it recoiled.
The reason for bringing this up is not so that you can plug numbers into a formulas in these exotic situations. The point is that the conservation laws have proven so sturdy exactly because they can easily be amended to fit new circumstances. Newton's laws are no longer at the center of the stage of physics because they did not have the same adaptability. More generally, the moral of this story is the provisional nature of scientific truth.
It should also be noted that conservation of momentum is not a consequence of Newton's laws, as is often asserted in textbooks. Newton's laws do not apply to light, and therefore could not possibly be used to prove anything about a concept as general as the conservation of momentum in its modern form.
Modern Changes in the Momentum Concept
Einstein played a role in two major changes in the momentum concept in the 1900's.
First Einstein showed that the equation p = mv would not work for a system containing objects moving at very high speeds relative to one another. He came up with a new equation, to which mv is only the low-velocity approximation.
The second change, and a far stranger one, was the realization that at the atomic level, motion is inescapably random. The electron in a hydrogen atom doesn't really orbit the nucleus, it forms a vague cloud around it. It might seem that this would prove nonconservation of momentum, but in fact the random wanderings of the proton are exactly coordinated with those of the electron so that the total momentum stays exactly constant. In an atom of lead, there are 82 electrons plus the nucleus, all changing their momenta randomly from moment to moment, but all coordinating mysteriously with each other to keep the vector sum constant. In the 1930s, Einstein pointed out that the theories of the atom then being developed would require this kind of spooky coordination, and used this as an argument that there was something physically unreasonable in the new ideas. Experiments, however, have shown that the spooky effects do happen, and Einstein's objections are remembered today only as a historical curiousity.
Momentum compared to kinetic energy
Momentum and kinetic energy are both measures of the quantity of motion, and a sideshow in the Newton-Leibnitz controversy over who invented calculus was an argument over whether mv (i.e., momentum) or mv2 (i.e., kinetic energy without the 1/2 in front) was the true measure of motion. The modern student can certainly be excused for wondering why we need both quantities, when their complementary nature was not evident to the greatest minds of the 1700's. The following table highlights their differences.
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