generally, if both bodies are moving, with different velocities, they simply exchange elocity at impact. Another example of an almost elastic interaction is magnetism. If we arrange a pair of U-shaped magnets in our glide blocks, so that they repel each other, hen one drifts quietly up to the other, it pushes it away and stands perfectly still, and now the other goes along, frictionlessly The principle of conservation of momentum is very useful, because it enables us to solve many problems without knowing the details. We did not know the details of the gas motions in the cap explosion, yet we could predict the velocities with which the bodies came apart, for example. Another interesting example is rocket propulsion. A rocket of large mass, M, ejects a small piece, of mass m, with a terrific velocity V relative to the rocket. After this the rocket, if it were originally standing still, will be moving with a small velocity, v. Using the principle of con- servation of momentum, we can calculate this velocity to be So long as material is being ejected, the rocket continues to pick up speed Rocket propulsion is essentially the same as the recoil of a gun: there is no need for any air to push against 10-5 Relativistic momentum In modern times the law of conservation of momentum has undergone certain modifications. However, the law is still true today, the modifications being mainly in the definitions of things. In the theory of relativity it turns out that we do have onservation of momentum; the particles have mass and the momentum is still given by mu, the mass times the velocity, but the mass changes with the velocity, hence the momentum also changes. The mass varies with velocity according to (10.7) where mo is the mass of the body at rest and c is the speed of light. It is easy to see from the formula that there is negligible difference between m and mo unless v is very large, and that for ordinary velocities the expression for momentum reduces to the old formula The components of momentum for a single particle are written as Py moy_, pz (10.8) wherev2=v2+Ux+v2. If the x-components are summed over all the inter acting particles, both before and after a collision, the sums are equal; that is, momentum is conserved in the x-direction. The same holds true in any direction In Chapter 4 we saw that the law of conservation of energy is not valid unle we recognize that energy appears in different forms, electrical energy, mechanical energy, radiant energy, heat energy, and so on. In some of these cases, heat energy for example, the energy might be said to be"hidden. " This example might suggest the question, "Are there also hidden forms of momentum-perhaps heat momen- tum? The answer is that it is very hard to hide momentum for the following The random motions of the atoms of a body furnish a measure of heat energ. 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 s 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