VELOCITY (Lat. velar, swift) is the Common term employed to denote speed, or rate of motion. it is obviously greater the greater the space passed over in a given time.
But, for its accurate measurement, we must distinguish between uniform and varying velocity.
Nothing is easier than the measurement of uniform velocity. It is measured by tha space passed over in a unit of time. Thus, we speak of velocities of 10 ft. per second, 20 m. per hour, etc. But, for scientific purposes, it is best to keep, as far as possible, to definite units of time and space; and those most generally convenient are the second and the foot. The latter is defined, from the imperial yard, by act of parliament: the former is usually chosen as the interval between the beats of a good mean-time clock. tinfortu uately, its duration is not invariable; but, as ages must elapse before any sensible alter ation takes place in its length, it may be used without inconvenient. lf, then, a be the velocity of a point moving uniformly, we mean that a feet are passed over in each second, so that, if 8 represent the space passed over in t seconds, we have a = at, a formula which contains the whole properties of uniform motion. It gives a = • that is, to find the velocity of a moving point (when uniform), divide the space (in feet) described in any period of time by the number of seconds in the period. This will give the same result whether we take a million seconds or the millionth part of a second, as the period in question. This at once shows us how to proceed in measuring a variable velocity, such as that of a stone let fall, in which case the velocity constantly increases, or of a stone thrown upward, in which case the velocity constantly diminishes.
That a moving body has at every instant, however irregular its motion may be, a definite velocity, is obvious, and is in fact matter of every-day remark. Thus when traveling in a railway train we say, shortly after starting: " We are now going at the rate of a mile an hour;" not thereby meaning that it will take us an hour to complete the mile, but that if we were to go on for an hour with the velocity we now have we should run a mile. Again we may say: " Now we are going at 30 m. an hour;" net thereby mean ing that we have so much as 30 in. to travel, or that our journey is to last more than perhaps a few minutes, but that an hour at the present rate would take us 30 miles. In common language, then, our question is how to measure our present rate.
If we could at any instant so adjust the steam-power to the resistance of the air and the friction of the rails as to keep the rate unaltered, we should have uniform velocity, measurable with ease, as above shown. But as we cannot generally do this (though Attwood's machine enables us to it in the case of a falling body), we are driven to some -other expedient. Now it is obvious that the smaller the interval we take the legs will
•our velocity have changed during its lapse, i.e., the more nearly will it have become uniform and measurable by the simple formula given above. That is for a variable velocity we have v = as an approximation, which is more and more nearly true as t, and therefore a, is smaller. In the language of the differential calculus—whose mental notions as laid down by its great inventor were in fact derived from this very question, the velocity being simply the fins-ion (q.v.) of the space described—we have it = Practically, by means of the electric chronoscope, we can now measure (very exactly) extremely small intervals of time, such, for instance, as the interval between the fall of the dog-head and the exit of the bullet from a rifle-barrel ; so that a variable velocity now presents no formidable difficulty, as we can study and measure it while it is almost absolutely uniform.
We define average velocity as the space described in any time divided by the number of seconds employed. This may not, except at one or more instants during the motion, represent the actual velocity; but it is a velocity with which, if uniform, the same space would have been described in the same time. We shall presently have an opportunity of usefully applying this definition to one interesting case of varying velocity.
The resolution and composition of simultaneous velocities follows, almost intuitively, from the most elementary geometrical notions. When a man is walking n.e. at a uni form rate, it is obvious to common sense that he is progressing northward and also eastward. What is his northward, and what is his eastward velocity. The 'answer is very simple. Suppose that in one second he walks from A to B, then AB represents his whole velocity. But draw AN northward and AE eastward; also draw BC parallel to AN.
Then AC is the space by which B is eastward of A, BC the space by which it is northward. Hence AC represents the east ward and CB the northward velocity (each being the space in its respective direction described in one second, and• these are called components of the velocity AB. AB again is said to be resolved into AC and CB.
The general proposition is this, that a velocity represented •by one side of a triangle may be resolved into two, represented in magnitude and direction by the other sides of the triangle. One or both of these may be again resolved by a similar process; and we find, as the most general propositions on the subject, that velocities represented by all the sides of a polygon (whether in one plane or not)