requisite to raise a weight w to a height h, is represented by wh; this then is the number of units of work accumulated in the body at the instant of projection. But since h is the height through which the body must fall to acquire the velocity v, therefore v2=2gh (Art. 47.); therefore h=1; whence g it follows that if U represent the number of units of work accumulated, Moreover it appears by the last article that this expression represents the work accumulated in a body weighing w pounds, and moving with a velocity of v feet, whatever may have been the circumstances under which that velocity was accumulated. The product() v' is called the VIS VIVA of the body, so that the accumulated work is represented by half the vis is called the MASS of the body.* viva, the quotient (2) 67. To estimate the work accumulated in a body, or lost by it, as it passes from one velocity to another. In a body whose weight is w, and which moves with a velocity there is accumulated a number of units of work ข there го represented (Art. 66.) by the formula. After it has passed from this velocity to another V, there will be accumulated in it a number of units of work, represented by V', so that if its last velocity be greater than the first, there will have been added to the work accumulated in it a num ber of units represented by V-; or if the second го g velocity be less than the first, there will have been taken from the work accumulated in it a number of units repre sented by — { ~ V 3. So that generally if U represent g g the work accumulated or lost by the body, in passing from the velocity v to the velocity V, then *Note () Ed. App. where the sign is to be taken according as the motion is accelerated or retarded. 68. The work accumulated in a body, whose motion is accele rated through any given space by given forces is equal to the work which it would be necessary to do upon the body to cause it to move back again through the same space when acted upon by the same forces. For it is evident that if with the velocity which a body has acquired through any space AB by the action of any forces whose direction is from A towards B, it be projected back again from B towards A, then as it returns through each successive small part or element of its path, it will be retarded by precisely the same forces as those by which it was accelerated when it before passed through it; so that it will, in returning through each such element, lose the same portion of its velocity as before it gained there; and when at length it has traversed the whole distance BA, and reached the point A, it will have lost between B and A a velocity, and therefore an amount of work (Art. 67.), precisely equal to that which before it gained between A and B. Now the work lost between B and A is the work necessary to overcome the resistances opposed to the motion through BA. The work accumulated from A to B is therefore equal to the work which would be necessary to overcome the resistances between B and A, or which would be necessary to move the body from a state of rest, and with a uniform motion, in opposition to these resistances, through BA. Let this work be represented by U; also let v be the velocity with which the body started from A, and V that which it has acquired at B. Then will (V2 —v3) repre W sent the work accumulated between A and B, ::. }', (V2—v')=U, .. V'—v2= W 2qU If the body, instead of being accelerated, had been retarded, then the work lost being that expended in overcoming the retarding forces, is evidently that necessary to move the body uniformly in opposition to these retarding forces through AB; so that if this force be represented by W U, then, since (V) is in this case the work lost, we where the sign is to be taken according as the motion is accelerated or retarded. 69. The work accumulated in a body which has moved through any space acted upon by any force, is equal to the excess of the work which has been done upon it by those forces which tend to accelerate its motion above that which has been done upon it by those which tend to retard its motion. For let R be the single force which would at any point P (see last fig.) be necessary to move the body back again through an exceeding small element of the same path (the other forces impressed upon it remaining as before); then it follows by Art. 54. that the work of R over this element of the path is equal to the excess of the work over that element of the forces which are impressed upon the body in the direction of its motion above the work of those impressed in the opposite direction. Now this is true at every point of the path; therefore the whole work of the force R necessary to move the body back again from B to A is equal to the excess of the work done upon it, by the impressed forces in the direction of its motion, above the work done upon it by them in a direction opposed to its motion; whence also it follows, by the last proposition, that the accumulated work is equal to this excess. Therefore, &c. *70. If P represent the force in the direction of the motion which at a given distance S, measured along the path, acts to accelerate the motion of the body, this force being understood not to be counteracted by any other, or to be the surplus force in the direction of the motion over and above any resistance opposed to it, then will P/S be the work which must be done in an opposite direction to over S come this force through the space S, or U=PaS, 71. If the force P tends at first towards the direction in which the body moves, so as to accelerate the motion, and if after a certain space has been described it changes its direction so as to retard the motion, and U, represent the value of U in respect to the former motion, and V, the velocity acquired when that motion has terminated, whilst U, is the value of U in respect to the second or retarded motion, and if v be the initial and V the ultimate or actual velocity, then As U, increases, the actual velocity V of the body continually diminishes; and when at length U, U,, that is when the whole work done (above the resistances) in a direction opposite to the motion, comes to equal that done, before, in the direction of the motion, then Vv, or the velocity of the body returns again to that which it had when the force P began to act upon it. This is that general case of reciprocating motion which is so frequently presented in the combinations of machinery, and of which the crank motion is a remarkable example. *72. If the force which accelerates the body's motion act always towards the same centre S, and Sb be taken equal to SB, it has been shown (Art 55.) that the work necessary to move the body along the curve from B to A, is equal to that which would be necessary to move it through the straight line A. The accumulated work is therefore equal to that necessary to move the body through the difference A of the two distances SA and SB (Art. 68.). If these distances be represented by R, and R., and P represent the pressure with which the body's motion along A would be resisted at any distance R from the point S, R1 then P&R will represent this work. Moreover the work accumulated in the body between A and B is represented W by (V-v), if V represent the velocity at B and v that 73. The work accumulated in the body while it descends the curve AB, is the same as that which it would acquire in falling directly towards S through the distance Ab, for both of these are equal to the work which would be necessary to raise the body from 6 to A. Since then the work accumulated by the body through AB is equal to that which it would accumulate if it fell through Ab, it follows that velocity acquired by it in falling, from rest, through AB is equal to that which it would acquire in falling through Ab. For if V represent the velocity acquired in the one case, and V, that in the other, then the accumulated work in the first case W W is represented by therefore W 9 V, and that in the second case by V,, W From this it follows, that if a body descend, being projected obliquely into free space, or sliding from rest upon any curved surface or inclined plane, and be acted upon only by the force of gravity (that is, subject to no friction or resistance of the air or other retarding cause), then the velo |