u2′ − u1′ = −(u2 − u1),
(40)
i.e. the relative velocity of the two bodies is reversed in direction, but unaltered in magnitude. This appears to be the case very approximately with steel or glass balls; generally, however, there is some appreciable loss of apparent energy; this is accounted for by vibrations produced in the balls and imperfect elasticity of the materials. The usual empirical assumption is that
u2′ − u1′ = −e (u2 − u1),
(41)
where e is a proper fraction which is constant for the same two bodies. It follows from the formula § 15 (10) for the internal kinetic energy of a system of particles that as a result of the impact this energy is diminished by the amount
| 1⁄2 (1 − e2) | m1m2 | (u1 − u2)2. |
| m1 + m2 |
(42)
The further theoretical discussion of the subject belongs to [Elasticity].
This is perhaps the most suitable place for a few remarks on the theory of “dimensions.” (See also [Units, Dimensions of].) In any absolute system of dynamical measurement the fundamental units are those of mass, length and time; we may denote them by the symbols M, L, T, respectively. They may be chosen quite arbitrarily, e.g. on the C.G.S. system they are the gramme, centimetre and second. All other units are derived from these. Thus the unit of velocity is that of a point describing the unit of length in the unit of time; it may be denoted by LT−1, this symbol indicating that the magnitude of the unit in question varies directly as the unit of length and inversely as the unit of time. The unit of acceleration is the acceleration of a point which gains unit velocity in unit time; it is accordingly denoted by LT−2. The unit of momentum is MLT−1; the unit force generates unit momentum in unit time and is therefore denoted by MLT−2. The unit of work on the same principles is ML2T−2, and it is to be noticed that this is identical with the unit of kinetic energy. Some of these derivative units have special names assigned to them; thus on the C.G.S. system the unit of force is called the dyne, and the unit of work or energy the erg. The number which expresses a physical quantity of any particular kind will of course vary inversely as the magnitude of the corresponding unit. In any general dynamical equation the dimensions of each term in the fundamental units must be the same, for a change of units would otherwise alter the various terms in different ratios. This principle is often useful as a check on the accuracy of an equation.