ON COLLISION. 7g 



since the height of the body's vertical ascent is in proportion to it; and some 

 liave considered it as the true measure of the quantity of motion ; but although 

 this opinion has been very universally rejected, yet the force thus estimated 

 well deserves a distinct denomination. After the considerations and demon- 

 strations which have been premised on the subject of forces, there can be no 

 reasonable doubt with respect to the true measure of motion ; nor can there 

 be much hesitation in alloAving at once that since the same force, continued 

 for a double time, is known to produce a double velocity, a double force 

 must also produce a double velocity in the same time. Notwithstanding the 

 simplicity of this view of the subject, Leibnitz, Smeaton, antl many others, 

 have chosen to estimate the force of a moving body, by the product of its 

 mass into the square of its velocity; and though we cannot admit that this 

 estimation of force is just, yet it may be allowed that many of the sensible 

 eft'ects of motion, and even the advantage of any mechanical power, however 

 it may be employed, are usually proportional to this product, or to the 

 weight of the moving body, multiplied by the height from which it must 

 have fallen, in order to acquire the given velocity. Thus a bullet, moving 

 with a double velocity, will penetrate to a quadruple depth in clay or tallow: 

 a ball of etjual size, but of one fourth of the weight, moving with a double 

 velocity, will penetrate to an equal depth: and, with a smaller quantity of 

 motion, will make an equal excavation in a shorter time. This appears at 

 first sight somewhat paradoxical : but, on the other hand, we are to consider 

 the resistance of the clay or tallow as a uniformly retarding force, and it will 

 be obvious, that the motion, which it can destroy in a short time, must be 

 less than that which requires a longer time for its destruction. Thus also " 

 when the resistance, opposed by any body to a force tending to break it, is to 

 be overcome, the space through which it may be bent, before it breaks, being- 

 given, Jis well, as: the force exerted aticvery point of that space, the power of 

 any body to break it is proportional to the energy of its motion, or to its 

 weight multiplied by the square of its velocity. 



In almost all cases of the forces employed in practical mechanics, the labour 

 expended in producing any motion, is proportional, not to the momentum, but 

 to the energy which is obtained ; since these forces are seldom to be considered 

 as uniformly accelerating forces, but generally act at some disadvantage, 

 when the velocity is already considerable. For. instance, if it be necessary to 



