CHAPTER V MOTION OF TRANSLATION 30. Velocity, Speed. The rate at which a point changes its position is called its velocity. From what has been said in § 27 it is evident that the change in the position of a particle must not only have magnitude, ie. there must be a certain distance measured along the path traversed by the particle between its first and last positions, but also the motion of the particle must have been in some direction, although not necessarily along a straight line, so that velocity is a vector. Velocity, therefore, may vary both in regard to its magnitude and also in regard to its direction. This may be illustrated by the motion of a train going round a curve. Here, although the magnitude of the velocity may be constant, ie. the train may travel along the rails for equal distances in each successive second, yet the direction of the motion is continually varying, since at any given point it is along the tangent to the curve at that point. Hence, to measure the velocity of a particle two things have to be determined: (1) the space which the particle has moved over in a given time, and (2) the change in the direction of motion during this time. In ordinary language, and in very many books on mechanics, the word velocity is used to indicate the first of these rates, i.e. the space passed over in a given time, without taking any account of any change in direction which may take place. Thus the end of the hand of a watch is said to move with uniform (i.e. constant) velocity, since it moves over equal spaces in successive equal times. It is, however, evident that the direction of the velocity is continually altering, and hence from this point of view the velocity is variable. It therefore saves confusion if we use, at any rate wherever ambiguity may arise, a separate word to denote the rate at which a particle describes its path, without reference to the direction, and for this purpose the word speed is generally used. Hence, if a particle moves in a straight line (so that the direction of motion remains constant), and passes over equal spaces in successive equal times, its velocity is said to be constant. If, however, a particle moves in a curve, so that its direction of motion continually changes, but passes over equal 1 The ratio of the total change in any quantity which occurs during a given time to that time is called the rate at which that quantity is changing. lengths of its path in successive equal times, its speed is said to be constant or uniform. Constant speed is measured by dividing the space passed over in any given time by that time. Thus, suppose a particle passes over a space, s, in a time, t, then the speed, v, is given by the equation v=s/t. Hence unit speed is such that unit space is passed over in unit time. In the cg.s. system unit speed is such that one centimetre is passed over in one second. The unit of speed has not received any recognised name, but when a particle passes over say 10 cm. in every second it is said to have a speed of 10 centimetres per second. This is often written 10 cm./sec. The only speed which has a recognised name is that of one nautical mile per hour, which is called a knot. The dimensions of speed can be obtained from the equation v=s/t by writing in the symbols for the units, when we get v[V]=s[L]] t[T]. If s and are each unity, then v is also unity, and we get the dimensional equation from which it follows that the dimensions of a speed are [LT-1]. The speed, v, is considered to be positive if the particle is moving away from the origin from which the distances are measured, and negative if the particle is moving towards the origin. Thus, if we measure vertical distances from the surface of the earth, the speed of a balloon is positive when it is ascending and negative when it is descending, while a bucket being lowered down a well has positive speed, but when being raised it has negative speed. 31. Variable Speed.-If a particle moves over unequal spaces in successive equal intervals of time, its speed is variable. Variable speed, at any instant, is measured by dividing the space passed over in a time, including the given instant, so small that during this interval the speed does not appreciably alter, by this interval of time. Suppose that during the very small interval of time which we may indicate by dt1 the particle 1 We here use of as a convenient symbol for a very small interval of time, or in other words, the symbol is used to indicate a small increment in the quantity to which it is prefixed. The expression of must not be looked upon as the product of ô and t, but as a single expression, in fact a kind of shorthand expressing a small increment of time. In the same way as represents a small increment in the length, measured from some fixed point, of the path traversed by the particle. Suppose, then. that during the time the particle has passed over a distance s, and at the time t+t it has passed over the space s +ôs, it is evident that it has passed over the space is in the time of, and its speed is δι moves over the space is, then the speed with which the body is moving at the time when the observation is made is given by It is important to be quite clear as to exactly what the above equation implies. In the first place it follows, from the definition of uniform speed given in § 30, that if a particle passes over a distance s in a time t, then the speed is given by st, whatever the value of t may be. Thus if a particle moving with a uniform speed pass over a metre in a second, the speed is one metre per second. It will pass over a tenth of a metre in a tenth of a second; the speed, however, will be the same (ie. } = I m sec.). In the same way it will pass over a millionth of a metre in a millionth of a second, and the speed will be as before, I metre I 10 = per sec. space passed over in this small interval of time, we should obtain the same value for the speed as would be obtained if we measured the space passed over in one second or a thousand seconds. Thus the value obtained for the variable speed (v=ds/dt) is such that if we had a particle moving with a constant speed, such that it passed over the space ds in the time t, it would in one second traverse a space equal to v units of length. We might therefore say that the variable speed of a particle is measured at any instant by the space passed over in one second by a second particle moving uniformly with a speed equal to that with which the first particle is moving at the given instant. (1). Hence if we were able to measure or calculate the A consideration of these two definitions will assist in making the matter clear; any difficulty which may be encountered may be lessened by recollecting that every one probably has some idea what they mean by saying that at any instant a train is travelling at, say, fifty miles per hour, though probably the train may not actually travel more than a mile or so in all. 32. Acceleration. The word acceleration is, in its most general sense, used to indicate any change in velocity. Hence it may mean an increase or a decrease in the speed, or a change in the direction of motion. In this sense acceleration is clearly a vector. It is, however, common to use the term acceleration with reference to the change in speed only, when it is a scalar. Acceleration may be uniform or variable. In uniform acceleration equal changes of speed occur in equal times. If the speed is increasing, then the acceleration is positive, while if the speed is decreasing, the acceleration is negative. Hence negative acceleration is what in ordinary language is called a retardation. Uniform acceleration is measured by the change in speed that takes place in a given time divided by that time. Hence if the speed of a particle change during the time from 7 to V, the acceleration given by the equation If the change in speed is unity, and takes place in unit time, we unit acceleration. Thus if the speed of a particle increase in one se by one centimetre per second, its motion has unit acceleration. T the unit of acceleration in the c.g.s. system. If the change of spe one second is r centimetres per second, then the acceleration is r metres per second per second,1 which may be written x cm/sec.2. Suppose the change in speed in a time to be v, then the acceler is given by a=v't If in this equation we introduce symbols for the units, we get a[A]=v[V]÷{[T]. Hence making a, v, and t each unity, we get the dimensional equati [4]=[V]÷[7]. Substituting the dimensions of [V] from § 30, we get The dimensions of acceleration are therefore I as regards length - 2 as regards time, and we are reminded of this double appearance to the second power) of the unit of time by the expression centimetre second per second. As in the case of variable velocity, variable acceleration is meas by dividing the change in velocity occurring in a time so small that acceleration does not appreciably change during the interval by time. 33. Velocity Curve.-Take two axes at right angles, ox, oy ( 11), and divide ox into a number of equal parts, OM1, M1 M2 MM3, and suppose each of these parts to represent an equal interval of ti say one second, so that we are measuring time along the axis o Next suppose we have a particle which starts from rest, and at the po O, M1, M., &c., we draw perpendiculars MP1, MP, &c., to represent speed of the particle at the instants of time represented by these poi i.e. at the commencement of the first, second, third, &c., second of mot Since the particle starts from rest, the perpendicular at O is zero. If the interval of time taken (i.e. a second in the above example) is sufficie small so that the velocity has not greatly changed from one point to 1 An acceleration is sometimes referred to as of so many centimetres per seco or feet per second. This is wrong, since acceleration is a change in speed and no position, as these expressions would lead us to suppose. next, the straight lines joining the points O, P1, P2, P3, &c., wi!l form a continuous curve; and this curve is called the velocity 1 curve. 1 Having drawn this curve, if through any point M, corresponding to a time, we draw a perpendicular to meet the curve at P, then MP represents the speed of the body at a time ʼn from the start. Speed If the speed of the particle is uniformly accelerated, the speed will increase by an equal amount in each unit of time. Thus, suppose that the particle starts from rest and moves with an acceleration a. Let OM' (Fig. 12) represent one second, then, since the speed of the particle at the end of a second is a, the ordinate M'p' representing the velocity at one second from the start is equal to a. If OM represents a time t, then, as the speed increases by an M Time- amount a in each second, the speed represented by MP will be at. join op and OP, since 'M' PM If we and the angles p'M'o and PMO are both right angles, it follows that the triangles 'OM' and POM are similar, and the angles poм' and POM are equal. The point must therefore lie on the straight line OP. In the same way it may be shown that all the extremities of 1 Strictly it ought to be called the speed curve. |