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Title: Kinetic Energy
Original Title: Force vive
Volume and Page: Vol. 7 (1757), pp. 112–114
Author: Jean Le Rond D'Alembert
Translator: John S.D. Glaus [The Euler Society,]
Original Version (ARTFL): Link

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Citation (MLA): D'Alembert, Jean Le Rond. "Kinetic Energy." The Encyclopedia of Diderot & d'Alembert Collaborative Translation Project. Translated by John S.D. Glaus. Ann Arbor: Michigan Publishing, University of Michigan Library, 2006. Web. [fill in today's date in the form 18 Apr. 2009 and remove square brackets]. <>. Trans. of "Force vive," Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers, vol. 7. Paris, 1757.
Citation (Chicago): D'Alembert, Jean Le Rond. "Kinetic Energy." The Encyclopedia of Diderot & d'Alembert Collaborative Translation Project. Translated by John S.D. Glaus. Ann Arbor: Michigan Publishing, University of Michigan Library, 2006. (accessed [fill in today's date in the form April 18, 2009 and remove square brackets]). Originally published as "Force vive," Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers, 7:112–114 (Paris, 1757).

Kinetic energy or force of bodies in motion is a term that was conceived by Mr. Leibniz so as to distinguish the force of a body actually in motion as opposed to the force of a body which only has a tendency to motion without actually being in motion; which must be discussed in further detail.

Let us suppose says Mr. Leibniz, that a weighted body is applied to a horizontal plane. This body attempts to drop but the effort is continually stopped by the resitance provided by the plane, in such a way this force is reduced to a simple tendency to motion. Mr. Leibniz called this force and the others of the same nature, dead forces.

Let us imagine the opposite, the same philosopher adds, that a body having weight which is thrown from up down and which in rising again continuously slows its motion due to the action of gravity, until that such time arrives and the force is totally lost, which is what happens when it achieves the highest point of its ascent; it is apparent that the force of this body vanishes by degrees and that it is in this way that its force is consumed. Mr. Leibniz calls this latter force, kinetic energy so as to distinguish it from the former, which grows and vanishes simultaneously; and in general if names kinetic energy that force of a body which moves from a continuously slowed and impeded by obstacles until finally this movement is entirely destroyed after having been successively diminished by minute degrees. Mr. Leibniz agrees that inactive force is similar to the mass times virtual speed, that is to say the speed with which a body tends to move following the general opinion. So that two bodies which collide or which pull on themselves directly and come into equilibrium, the product of the mass by speed is virtually the same in either way. However in this case, the force of each of these two bodies is potential force , since it has been stopped altogether and as it is a whole by a contrary force. Therefore in this case, the product of the mass by the speed must represent force. However Mr. Leibniz holds that kinetic energy must be measured differently and that it is as the product of the mass times the square of the speed; that is to say that a body which contain a certain force when it moves with a given speed will have a quadrupled force , and if it moves with a doubled speed; a force nine times larger, if moving with a speed three times faster etc., and that in general if the speeds are successively 1,2,3,4 etc., the force will be as 1,4,9,16, etc., that is as the squares of the numbers 1,2,3,4, as though the body were not really in motion but tended to move with the speeds 1,2,3,4 etc., its force would be a potential force would then be as 1,2,3,4, etc,.

In a contrary system to kinetic energy, the force exerted on a body in motion is always proportional to that which is called otherwise quantity of motion, that is to say the product of the body’s mass multiplied by the speed, instead of that which it is in the contrary system, it is the product of the quantity of motion by the speed.

To reduce this question into its simplest terms, it requires knowing if the body’s force which has a certain speed, if it increases twice or four times when its speed doubles. All experts in Mechanics had believed until Mr. Leibniz that it was simply doubled, this great philosopher was the first to uphold the former that it was four times and he proved it with the following reasoning. The force of a body can only be measured by its effects and by the obstacles it overcomes. If a weighted body being thrown upwards with a certain speed achieves a height of fifteen feet, everyone is convinced that it will achieve a height of sixty feet having been thrown with twice the speed, See Acceleration. In this latter case the effect is four times greater than the first. In his work , Discourse on communication within the laws of motion published in 1726, which Mr. Johann Bernoulli appended into the general collection of his works by adding Mr. Leibniz’ proof and great number of other proofs. He has shown that a body that compresses or releases, as does a spring with a certain speed can close four springs twice as fast similar to the first and nine springs by three times the speed etc.Mr. Bernoulli strengthens this novel argument in favor of kinetic energy with other highly interesting and important observations, that we will have the opportunity to discuss later in the article on the Conservation of kinetic energy. At one time this work produced a schism between scientists concerning the measurement of forces.

The principle answer that has been made to the partisan objections of kinetic energy, see the mémoires of the Academy 1728, consist in its uniformly slowed motion and to recognize in this case that force is not similar to speed: it is alleged that a body which travels fifteen feet from down to up, will travel sixty feet with twice the speed. It would have travelled these sixty feet in a time double of the first. If its motion were uniform, it would have travelled in this a same time one hundred and twenty feet, See Acceleration. However in the case where it would travel fifteen feet in a delayed action, it would travel thirty feet in the same time and sixty feet in double the time with uniform motion. It produces the effect as 120 to 60 or 2:1 and consequentially the force in the first is nothing but twice the other and not quadrupled. Thus it can be concluded, that a falling body covers four times more space with twice the speed, but that it does this in twice the time. And that this is the equivalent to twice the effect rather than four times. It is said that one must divide the space by the time to have the effect to which the force is proportional and not to make the force proportional to the space. The proponents of kinetic energy by saying that the nature of force which is greater is to last longer and thus it is not surprising that a heavy body which covers four times more space, will do so in twice the time , and the true effect of force is to allow it to cover four times the space; and that the more or the less time is of no consequence since that the more or the less time comes for the magnitude of the force and that it is not true to say as it will somehow produce the responses from their adversaries and that is that the smaller the force, all things which are equal, then the times will be greater, since it appears to the contrary it is infinitely more natural to believe that it should be as large as it requires that much more time to consume itself.

Finally, it is important to note that to suppose that force is proportional to the square of its speed, it is not necessary, according to those proponents of kinetic energy, that the force is self-consuming both in reality and actually through its action; it is sufficient to imagine that it can be used up and dissipated little by little by infinitely small degrees. In a body which moves uniformly, force is not less proportional to the square of the speed according to these Philosophers, even though this force remains the same since that if this force was exercised against an obstacle which would dissipate it by degrees, its effect would be similar to the square of the speed.

We send our readers to what has been written for and against live forces in the 1728 mémoires of the Academy of Saint Petersburg, Volume 1 and in other works. However instead of bringing up all that has been said on this matter, it will not be a useless thing to explain succinctly those principles which might serve to solve it.

We one speaks of the force of bodies in motion, where one provides only lip service to the word without being aware of its meaning, from which one can only understand by this in general the properties that bodies in motion possess, to overcome the obstacles that they encounter or to resist them. Thus it is neither through space that a body in which a body travels uniformly nor by its time that it uses to travel, or by the simple and unique and abstract though of its mass and its speed that immediately comes to mind to estimate force. It is uniquely by the obstacles which a body encounters and by the resistance that are provided by these obstacles. The greater that a body can overcome an obstacle or that it can resist greater will be its force, as long as not wanting to represent by this word the implication that a presumed entity resides in this body that one does not use it in some abbreviated form to express a fact; such as is said that a body has twice the speed of another, in stead of saying that the body travels in equals times twice the distance without implying for this that the word speed represents an inherent being of the body.

With this understood, it is clear that one can oppose motion of a body with three separate obstacles; where there are obstacles which cannot be overcome and entirely inhibit the body’s motion no matter what it can be; or obstacles which contain precisely the amount of resistance to prohibit the motion within a body and that which destroys instantaneously which is in the case of equilibrium; or finally obstacles which prohibit motion a little at a time, which is in the case of slowed motion. As those obstacles which cannot be overcome and destroy equally all types of motion, they cannot serve any purpose in understanding force. Therefore it is only in equilibrium or in slowed motion that we must find out the measurement. Since everyone agrees that when the product of the mass multiplied by the virtual speed, that is by the speeds at which they tend to move are equal in part and other. Therefore in the equilibrium, the product of the mass by the speed, or the quantity of motion which is the same thing can represent force. Everyone is in agreement that when a body is in slowed motion, the number of obstacles which must be overcome is as the square of the speed in such a way that if a body should have closed a spring, for example with a certain speed could with a double speed close at the same time or successively, not two but four springs similar to the first, nine with a triple speed and so forth.

From which the partisans of kinetic energy conclude that the force of bodies which are actually in motion is in general similar to the product of the mass multiplied by the square of the speed. In the end what inconvenience could there be when the measurement of the forces is different in its equilibrium and it its slowed motion, since if we only want to rationalize on clearly presented ideas, we should hear nothing more from the word force than the effect produce in overcoming the obstacle, or by resisting it. One must however admit that the opinion of those who regard the force as the product of the mass times the speed, can equally take place not only in the case of equilibrium but also in that of slowed motion. Since the sum of resistance is proportional to the quantity of motion, since by all accounts the quantity of motion that a body loses at every moment is proportional to the product of the resistance of the extremely infinitesimal duration of the instant. And that the sum of these products is evidently the total resistance. All of the difficulty resides in that fact as to know whether one must measure the force by the absolute quantity of the obstacles, or by the sum of their resistance. It would appear more natural to measure the force in this latter way, since an obstacle is not unless it provides resistance and it is properly speaking the sum of the resistances which overcomes the obstacle. Furthermore by estimated the force in such a way there is the advantage of having both for the equilibrium and the slowed motion a common measure; however, since we neither have a precise and distinct idea concerning the word force, by retaining this term to express an effect, I believe it should be left to each to be the master and decide as he wishes to express it and the question can no longer consist except in a very useless metaphysical discussion or in a battle of words made even more corrosive with philosophers debating.

That which we have just said concerning the famous question of kinetic energy is taken from the preface of our treatise of Dynamics published in 1743 at a time when this question was still strongly disputed among scientists. It appears that geometers agree today nearly unanimously that which we have supported all along, that it is semantics and how in the world could it not be this since the two parties are in entire agreement concerning he fundamental principles of equilibrium and motion? In effect, we should propose a question in Dynamics to be resolved by two able geometers, for which one is against and one is for forces vives kinetic energy and as long as their solutions are correct they will be in agreement. The measure of these forces is also useless to mechanics as are those concerning space and motion and as to what we have said concerning this can be read under the word Elements of Science, volume V. page 493.col. 1and 2. There are only two things that we see clearly concerning a body’s motion; the distance travelled and the time that was used to travel that distance. It is only from this notion that one must deduce all Mechanical principles and from which they can be deduced. See Dynamics.

One point which should not be neglected and proves that it only is mention here of a point of identifying the definition of a name, and that is that a body has a tendency to motion but is impeded in doing so because of some obstacle, or that it moves due to a uniform motion with the speed at which this input dictates, or finally that in beginning to move with this speed its motion is impeded little by little by this obstacle, in any event, the effect which is produced by the body is different, however the body in itself is not in receipt of anything new, it happens that only its action is applied differently.. Therefore when it is said that a body’s force is similar in cases to its speed and at other times as the square of its speed one is attempting to say that only that the effect in certain cases is similar to the speed and in others similar to the square of the speed. One must also take note that the word effect is somewhat vague and it necessarily requires a better definition with more precision, since it possesses different meanings in each one of the three cases that we have just mentioned. In the first it means the effort that the body makes against the obstacle; in the second, the distance travelled in a given and constant time; in the third the total distance travelled until all motion has ceased without the least regard to the time in which the force has taken to consume itself.

Accordingly one should note by all that we have said concerning a body whose tendency to motion is applied differently produces different effects; some proportional to its speed, some to the square of its speed. By knowing this the axiom that the effects are proportional to their causes is at the very least very poorly defined, since here is a same cause which produces different effects. It is necessary to restrict this proposition to which it necessarily states that the effects are proportional to their cause acting in the same manner. However we have already made clear to the words Accelerator and Cause that this axiom is very vague and very poorly defined and absolutely useless to Mechanics and capable of leading to conclusions if not used with precaution.