194. =Relation of Force to Velocity.=--It would seem at first thought that the motion produced in any body must be in exact proportion to the force producing it; that is, that twice the force which produces a given velocity would double that velocity, and three times would treble it, etc. This is true where there are no obstacles to motion, as in the case of the heavenly bodies moving in their orbits. But in all motions here upon the earth there are obstacles; and as reaction is always equal to action, the greater the velocity the greater is the reaction of the obstacle. If, therefore, you increase the velocity of any body, you not only have to communicate more motion to it, but you must overcome also the increased reaction. The rate of increase of force for increased velocities has been very accurately ascertained. This I will explain. A boat moving from B to A, Fig. 130, we will suppose, displaces a quantity of water represented by the space between the two lines extending from B to A. Now if it move from B to C, it displaces twice the bulk of water B C; and as it is displaced in the same time that B A was, each particle is displaced with twice the velocity. Double the force is required to displace a double portion of water, and to do this with double the velocity the force must be doubled again. So if the boat is made to move three times as far in the same time, that is from B to D, three times the quantity of water is displaced, and each of these three portions, B A, A C, and C D, is displaced with three times the velocity. The force required, then, to do this is nine times that required to carry the boat from B to A in the same time. It is plain, therefore, that with velocities represented by the numbers 1, 2, 3, 4, etc., the forces requisite to produce these velocities must be as the squares of these numbers; viz., 1, 4, 9, 16, etc. This law is a very important one in a practical point of view. For example, it shows us how much larger a quantity of coal is required to produce in steamboats a high velocity than a moderate one. Its application too to the science of gunnery is important.