§ 12. Rectilinear Motion.—Let x denote the distance OP of a moving point P at time t from a fixed origin O on the line of motion, this distance being reckoned positive or negative according as it lies to one side or the other of O. At time t + δt let the point be at Q, and let OQ = x + δx. The mean velocity of the point in the interval δt is δx/δt. The limiting value of this when δt is infinitely small, viz. dx/dt, is adopted as the definition of the velocity at the instant t. Again, let u be the velocity at time t, u + δu that at time t + δt. The mean rate of increase of velocity, or the mean acceleration, in the interval δt is then δu/δt. The limiting value of this when δt is infinitely small, viz., du/dt, is adopted as the definition of the acceleration at the instant t. Since u = dx/dt, the acceleration is also denoted by d2x/dt2. It is often convenient to use the “fluxional” notation for differential coefficients with respect to time; thus the velocity may be represented by ẋ and the acceleration by u̇ or ẍ. There is another formula for the acceleration, in which u is regarded as a function of the position; thus du/dt = (du/dx) (dx/dt) = u(du/dx). The relation between x and t in any particular case may be illustrated by means of a curve constructed with t as abscissa and x as ordinate. This is called the curve of positions or space-time curve; its gradient represents the velocity. Such curves are often traced mechanically in acoustical and other experiments. A, curve with t as abscissa and u as ordinate is called the curve of velocities or velocity-time curve. Its gradient represents the acceleration, and the area (∫u dt) included between any two ordinates represents the space described in the interval between the corresponding instants (see fig. 62).
So far nothing has been said about the measurement of time. From the purely kinematic point of view, the t of our formulae may be any continuous independent variable, suggested (it may be) by some physical process. But from the dynamical standpoint it is obvious that equations which represent the facts correctly on one system of time-measurement might become seriously defective on another. It is found that for almost all purposes a system of measurement based ultimately on the earth’s rotation is perfectly adequate. It is only when we come to consider such delicate questions as the influence of tidal friction that other standards become necessary.
The most important conception in kinetics is that of “inertia.” It is a matter of ordinary observation that different bodies acted on by the same force, or what is judged to be the same force, undergo different changes of velocity in equal times. In our ideal representation of natural phenomena this is allowed for by endowing each material particle with a suitable mass or inertia-coefficient m. The product mu of the mass into the velocity is called the momentum or (in Newton’s phrase) the quantity of motion. On the Newtonian system the motion of a particle entirely uninfluenced by other bodies, when referred to a suitable base, would be rectilinear, with constant velocity. If the velocity changes, this is attributed to the action of force; and if we agree to measure the force (X) by the rate of change of momentum which it produces, we have the equation
| d | (mu) = X. |
| dt |
(1)
From this point of view the equation is a mere truism, its real importance resting on the fact that by attributing suitable values to the masses m, and by making simple assumptions as to the value of X in each case, we are able to frame adequate representations of whole classes of phenomena as they actually occur. The question remains, of course, as to how far the measurement of force here implied is practically consistent with the gravitational method usually adopted in statics; this will be referred to presently.
The practical unit or standard of mass must, from the nature of the case, be the mass of some particular body, e.g. the imperial pound, or the kilogramme. In the “C.G.S.” system a subdivision of the latter, viz. the gramme, is adopted, and is associated with the centimetre as the unit of length, and the mean solar second as the unit of time. The unit of force implied in (1) is that which produces unit momentum in unit time. On the C.G.S. system it is that force which acting on one gramme for one second produces a velocity of one centimetre per second; this unit is known as the dyne. Units of this kind are called absolute on account of their fundamental and invariable character as contrasted with gravitational units, which (as we shall see presently) vary somewhat with the locality at which the measurements are supposed to be made.
If we integrate the equation (1) with respect to t between the limits t, t′ we obtain
mu′ − mu = ∫t′t X dt.
(2)