Speed. Change in motion by which a body is capable of traveling through a certain space in a certain time. See Motion.

Leibniz, Bernoulli, Wolff, and the other partisans of kinetic energy [ forces vives ] claim that one should estimate the force of a body in motion by the product of its mass times the square of its speed; those who have not accepted the sentiment of these savants wish that force be nothing else than the quantity of motion, or the product of the mass times the speed. See Kinetic energy.

The uniform speed is that which allows the vehicle to travel in equal spaces at equal times. See Uniform.

There is but one space which would offer no resistance, in which a perfectly uniform motion would be executable, in the same way that there is but one space in which perpetual motion would be possible; since in this space there would be nothing that it might encounter which would cause the motion of the body to accelerate or slow down. The inequality or the non uniformity of all the motions that we are aware of, is the counter-proof of perpetual mechanical motion which so many people have sought; it is impossible in view of the continuous loss of force that occurs to bodies in motion, due to the resistance of the media in which they travel, the friction of their parts, etc. Therefore, in order for a perpetual mechanical motion to occur it would be necessary to find a body that is without friction, or one which the Creator has endowed with an infinite force, by which it would overcome the constant repetitions happening at every moment. Furthermore, even though, properly speaking, there is no such thing as perfectly uniform motion, when a body moves within a space that provides little resistance and when the body is neither the recipient of acceleration nor of any noticeable decrease, one considers its motion as though it were perfectly uniform. [1]

Speed is considered to be either absolute or relative; the definition that we have provided corresponds to simple or absolute speed , that by which a certain space is crossed within a certain time.

The actual or absolute speed of a body is the relationship between the space that it traverses and the time in which it is in motion. The respective speed is that in which two bodies either approach or distance themselves from each other within a certain space and within a determined time, irrespective of their actual speeds . Thus, the absolute speed is a positive thing; however, the respective speed is but a simple comparison that the mind makes between two bodies, according to whether they approach or distance themselves from each other. [2]

The speed with which two bodies distance or approach each other is their relative or respective speed , whether each of these bodies is in motion, or only one of them. Even if one body is at rest, one can look at it as having a speed relative to another assumed to be in motion; if two bodies in one second find themselves closer than they were by two feet, their respective speed will be double that of two bodies that would in the same time have traveled only one foot towards each other, assuming the motion to be uniform.

A non-uniform speed is one which receives some increase or some decrease; a body has an accelerated speed when a new force acts upon it and increases its speed . For this effect to occur the new force which acts upon it, must act is whole or in part in the direction following the body in motion has already taken.

The speed of a body is slowed, when some force opposed to its own removes some of its speed .

The speed of a body is equally or unequally accelerated according to whether the new force which acts upon it does so equally or unequally in the same amount of time; and it is equally or unequally slowed according to whether the losses which have occurred are either equal or unequal in the same amount of time.

The speed of bodies traveling along curved lines . According to Galileo’s system concerning falling bodies (a system which is accepted throughout the world), the speed of a body falling vertically is at each moment of its fall proportional to the root of the height from which its has fallen. After having discovered this proposition Galileo also recognized that if the body fell along an inclined plane, the speed would be the same as though it had fallen from the vertical which was measured at the same height, and he extended the same conclusion with the construction of numerous inclined planes which made random angles, while always claiming that the speed at the end of the fall made along these different planes must be the same as if it had fallen vertically from the same height.

This last conclusion was accepted by all mathematicians, until 1693, when M. Varignon [3] demonstrated its error, in noting that the body which had just traversed the first inclined plane and which arrives on the second, strikes it with a part of the speed which is lost and which consequently stops it from being in the same situation had it fallen from a single inclined plane, which would not have had any fold. M. Varignon after having noticed this error, explained the matter in such a way as to prevent anyone falling into the opposite error, to which one was naturally drawn, which was to believe that the fall of a body along a curved line, that is to say, along an infinite number of inclined planes, would also be unable to produce the speeds equal to those of a body that had fallen vertically from the same height. In order to show the difference between these two cases, he showed that when the inclined planes constitute infinitely small angles as happens in curves, the lost speed at each of these angles, is an infinitely small curve of the second order, such that after an infinite number of these falls, that is, after the entire fall through the curve, the speed which is lost is an infinitely small amount of the first order, which can be ignored consequently when applied against a finite speed. One may also see more on this topic in our treatise on dynamics, part one towards the end. [4]

Just as an equation between two variables can express any random curve, for which the coordinates are the variables of this equation; one may also express by the variables of an equation the different speeds that two forces would produce separately in a similar body; and if these forces are supposed to act in parallel to the two lines whose positions are given, from which we suppose these variables are taken, the curve expressed by the equation will then be that which the body describes, in virtue of the two forces combined together. If, for example, one supposes that that one of these forces is gravity and that the other is nothing more that the first finite impulse from which there is no acceleration, the curve having proportional ordinates to the squares of the abscissas, will be a parabola. See Parabola.

In order to measure a random speed in a consistent way that will help to determine all other speeds, one takes the quotient of the distance by the time, supposing that this space is traversed by virtue of this supposed constant speed. If, for example, a body, at its actual speed could travel 80 feet in 40 seconds, one would have 80/40 or 2 to express its speed, in such a way that if we compared this speed to that of another body which did 90 feet in 3 seconds, we would find in the same way 90/3 or 3 as this new speed, we would recognize by this means that the relationship of these speeds is 2 to 3.

s generally being space, and t the time, st is the speed; insofar as the motion is uniform. One might make a fairly well-founded objection concerning the measurement of speed: one might say that space and time are two heterogeneous quantities which cannot be compared and that we do not have a clear idea of the quotient st; to which one must reply that this expression of speed does not mean anything, except that the speeds of the two bodies are always between themselves like the quotients of spaces divided by the times, insofar as we represent the spaces and the times by abstract numbers which have between them the same relationship as these spaces and these times. See the end of the article Equation.

If the motion is variable, it is supposed constant during the description of infinitely small ds of space and one then expresses the speed by ds , dt . See Motion.

1. This paragraph is attributed in the text to Formey.

2. This paragraph is attributed in the text to Formey.

3. Pierre Varignon (1654-1722), French mathematician.

4. The reference is to Jean Le Rond d’Alembert, Traité de dynamique, dans lequel les loix de l’équilibre et de mouvement des corps sont réduites au plus petit nombre possible (Paris, 1743). The reference confirms that d’Alembert is the primary author of the article, despite the fact that it is unsigned.