The times of the rise and the subsequent fall are equal.
II. Since the heights are as the squares of the times of the fall or ascent, we have 1" : and h= 16 and V h =4 t.
, h h also =—, and t 16" 4 A heavy body, falling during four seconds, falls 256 feet. A body rising straight upwards 144 feet employs 3 seconds in its ascent.
III. Because the heights fallen through are also propor tional to the squares of the velocities acquired at the end of the fall, we have =16:h 16 and h=7 , andvh=± 32 32 8 and, conversely, v =8 vh, and h.
All questions concerning the perpendicular ascents and descents of heavy bodies may be solved by means of the two equations An easy mode of extempore computations is had, by remarking, that since a heavy body falls 16 feet in a se cond, and acquires the velocity 32, it falls 1 foot in -1th of a second, and acquires the velocity 8.
In every second of the fall, the velocity is increased by 32 ; and in every foot of the fall, the square of the veloci ty is increased by 64.
In many questions, particularly in hydraulics, it is con venient to have the measures in inches.
Now, vr=386 : 27,785. Therefore a heavy body by falling one inch acquires the velocity 27,785 inches, or nearly 274 inches per second.
Did gravity impel a body uniformly along a space equal to the radius of the earth, it would generate the ve locity which would enable the body to describe a parabola, having the centre of the earth for its focus. If project ed straight upwards with this velocity it would never re turn.
Now V 16 : V Earth's rad.=32 : 36,680 feet. This is the velocity now spoken of. Suppose the earth uniformly dense, and a pit to the centre. A heavy body would ac quire, by falling down this pit, the velocity 25,866. Greater velocities than either of these can be produced by forces which we know. Aurum fulminans expands with the velocity of at least 12 miles per second.
It does not seem necessary to insist further on the rec tilineal ascents and descents of heavy bodies, and therefore we proceed to consider their curvilineal motions, when projected in any direction that deviates from the perpen dicular. These are the motions which are understood
to form what is called PROJECTILES.
These motions are not only interesting to the philoso phical mechanist, as examples of a constant deflecting force, and a uniform deflection in parallel lines, but also to the artillerist ; because the motion of shot and shells are cases of this question, which comprehend the whole of his art. It has therefore been very much cultivated ; and there is no branch of mechanical philosophy on which so much has been written, or so many experiments made for its improvement. The experimental cultivation of this branch could scarcely be prosecuted by private persons ; but, in all the states of Europe, there are public establish ments for this purpose, and no expence has been spared for bringing to perfection an art on which the fate of nations has unfortunately much dependence.
But, notwithstanding this liberal encouragement, and the numberless volumes which have been published on the subject, it cannot be said to have improved much as a sci ence since it came out of the hands of its inventor, and his immediate pupil Tartaglia ; and we shall be greatly disap pointed if we look for that nice agreement between the re sults of the most approved theory and what we observe in the flight of great shot and shells. The theory, how ever, is unexceptionable ; and the enormous deviations that we see in the actual performance of artillery, is owing to the resistance of the air. This was long considered as insignificant, even after Newton had given us sufficient in formation to the contrary. But the gentlemen of the pro fession made little account of the speculations of a private philosopher, and continued to regulate their theories by no tions of their own. They have been at last convinced of their mistake by the curious experiments and discoveries of Mr Robins, and are improving their practice in some measure. But we now find, that the theory of the motion of heavy bodies through a resisting fluid, is one of the most abstruse and difficult tasks that the mechanician can take in hand.