Mechanics

Distance Represented by Area Under Curve.-3. Mathemati cally, the formula (3) implies that i.e., the distance s may be found by integrating the speed S with respect to time t. Let S and t be related, after the manner of fig. i, in a speed-time diagram (fig. 2) : the graphical interpretation of (4) is that the area under the portion PQ of the speed-time dia gram measures (on an appro priate scale) the distance trav elled in the interval represented by LM.

To establish this rule independ ently, we have only to notice, (i) that it would be obviously true if the speed were uniform—in which case PQ would be a horizontal straight line, and the area would be proportional to (speed) X (length of interval) ; that it would be true if the speed changed suddenly at the ends of finite but short intervals, and remained constant during those intervals, so that the speed-time curve con sisted of a number of small steps, as indicated in fig. 2; and (3) that the conditions contemplated in (2) will be indistinguishable from the actual conditions, and the stepped diagram indistinguish able from the actual curve, if the intervals are sufficiently short.

Acceleration.-4.

When the speed varies, we use the term acceleration to denote the rate at which it increases. Thus, if the speed of a train changes from 6 to i o ft. per second in an interval of 2 seconds, the total increase in speed during this interval is 10-6 = 4 ft. per second, and hence the average rate of increase, i.e., the average acceleration of the train, is 4. = 2 ft. per second per second. In symbols, if the speed increases by S' in an interval of time t', then the average acceleration is measured by — • t' To obtain an accurate measure of varying acceleration, we make the time interval t', as before, indefinitely small. Corresponding to (3) we have, as an expression for the instantaneous accelera tion f, Thus the acceleration f may be found by differentiating the speed S, and the. speed by integrating the acceleration, with respect to the time t. Similarly, in relation to graphical methods, we may say that : (a) in a speed-time diagram, the acceleration corresponding to a point P is measured (on an appropriate scale) by the slope of the tangent at P; (b) The area under a portion PQ of the acceleration-time diagram measures (on an appropriate scale) the increase of speed in the corresponding interval of time.

Velocity.-5.

In this discussion of speed and acceleration, we have not been concerned to know the route of our train. Dis tance (s) has meant distance measured, from some arbitrary starting point, along the route ; and in this sense a train can be said to be distant s from the starting point, although it may be, if the route is circuitous, much nearer "as the crow flies." Speed and acceleration have been understood, similarly, as measured along the route. Direction has had no meaning, except in the sense that the train may be travelling forwards or back, i.e., with positive or negative speed.

These ideas were sufficient, because we were concerned with motion in some definite path, or route If, instead of the train, our concern had been with a ship, we should have had to consider the direction, as well as the speed, of its motion at every instant.

Whereas the position of the train could be specified by means of one quantity (the distance s), two quantities are required to fix the position of the ship: thus, if 0 (fig. 3) is a fixed point on the earth's surface, and ON, OE are lines through 0 in a northerly and easterly direction, the position P can be specified by assigning values to the distances PK, PL. Let QPR be the path of the ship. Then, as the ship moves from P in the direction of R, the dis tances PK, PL, i.e., the lengths OL and OK, will increase. Let x and y (according to the usual convention) denote these dis tances, or coordinates. Then the speed u with which the ship is travelling east will be the rate at which x increases; i.e., we shall have, as in equation (3) Composition and Resolution of Accelerations.-7. Similar relations hold for accelerations; i.e., a resultant or total accelera tion F may be resolved into two components (e.g., an eastward acceleration f, combined with a northward acceleration fy) which are related with F by the parallelogram law. Corresponding to equation (5), we shall have