TRANSLATION. In motion of translation it is necessary to consider the motion of one point of the figure only, as that is the same for all the points. If the figure is moved from one position to another. this displacement may be represented by a straight line joining the initial and final positions of any one point of the figure.
line indicates by its direction and its length the displacement of the whole figure; it is called :t rector, and displacement is said to he a vector quinifity because it requires for its complete un derstanding a direction and a numerical quantity only and so can he pictured by a straight line having the proper direction and a length equal to or proportional to the numerical quantity.
If the motion of the figure is Illliform—that is. if it passes over equal distanees in equal in tervals of time—the rate of motion, or the dis tame traversed divided by the time taken, is called the linen- spred. if the motion is not uni form. the linear speed at any instant is the dis. tame which the figure would move in the next second if its motion were to continue for that interval of time at exactly the same rate it is at that instant: in mathematical symbols. if Ar is the length of the extremely short distance traversed in the extremely short interval of time At immediately following the given instant, the linear speed at that instant is the value of At in the limit as At is taken smaller and smaller. Speed is therefore a number. If the speed in a particular direction is considered—that is. if a distinction is made between the motions of fig urcs with the same speed but in different diree tions—the linear speed in a given direction is called the linear velocity in that direction. Linear velocity is evidently a vector quantity; the linear speed giving the numerical quantityA i.e. the length of the vector.
If a figure is given simultaneously two dis placements, the resulting displacement is evi dently found by 'adding geometrically' the two components. Thus if All and BC represent the two component displacements, the actual one will be AC. formed by placing BC so as to con tinue the motion indicated by AB and completing the triangle. (A man walking across the deck of a moving ship illustrates this 'composition' of displacements.) Similarly, if AB and BC repre sent the linear ve locities of the two component motions, the actual velocity is represented in di rection and speed by AC. In a perfectly siturvhae manner, three, four. etc., vec tor qualities may be added geometrically. Further. conversely, any displacement or velocity may be re garded as made up of two displacements or two velocities, the condition being that the two vectors represent ing the component quantities should form a broken line joining the ends of the vector rep resenting the actual quantity. This is called
'resolution' of displacement or velocity. In re solving vectors it is nearly always best to take the components so that they are at right angles to each other, for then they arc independent of each other. Thus if All is a displacement—or any vector—its 'component in the direction' AF is the vector AC obtained by dropping a per pendicular from 11 npon AK. AB is equivalent to AC and CB, hut CIS has no connection with the direction AF, and AC is then that component of AB which indicates how much AB is con cerned with the direction AF. In mathematical language the component in the direction AF of a vector AB is All cos (('AB).
general the velocity of a moving figure will not be eonstant ; and the rate of change of the linear velocity at any instant—that is. if Ar is the extreuuel• small cluing,- of the vcloeity in the extremely small interval of time At, the limiting value of called the linear ac celeration at that instant. It is evident that acceleration !slug the change in velocity, and therefore the differenee between two lines, is it self a vector quantity: it has a numerical value and a definite direction, and as with displace ments and velocities, accelerations can be corn pounded by geometrical addition or resolved into components. Since linear velocity is character ized by a speed and by a direction, it can change in two independent ways: the speed can change, the direction remaining the same, e.g. a failing body; the direction can change. the speed remain ing the same, e.g. a particle moving in a circle at a uniform rate. (In general, both speed and direction change, e.g. a simple pendu lum.) There are therefore two types of linear acceleration. The three most interest ing cases of linear acceleration are the following: (1) Notion in a straight line, constant ac celeration. If the acceleration is positive, the speed increases; if it is negative, the speed de creases. Let the acceleration he called a, and the speed at any instant then, t seconds later, the speed will be s = at, and the distance traversed in that time will be x = + If t is eliminated from these equations it is seen that a' = 2ax. These formulae apply to a body falling freely toward the earth, in which case u = 9S0; to a body thrown vertically up ward. in which case a = — DSO; and to many other illustrations.