UNNERT. It ia intended therefore in this place only to consider the laws of the vertical ascent and descent of bodies in resisting media, the force of gravity, or of terrestrial attraction, being supposed to be constant; and in non-resisting media, under the condition that the force of gravity is variable.
Let a spherical body descend vertically from a state of rest in a resisting medium (air, water, fie.) supposed to be of uniform density ; and let it be admitted, agreeably to the Newtonian hypothesis Princip.', lib. ii. sec. I ; SchoL), that the resistance of the medium is proportional to the square of the velocity, v, acquired at any moment in the descent ; then, if we suppose U to be the velocity which a body falling towards the earth in the resisting medium would acquire when that resistance becomes equal to the accelerative force of gravity, the latter being, as usual, represented by g feet), we shall have : g : g ; and the lard term represents the resistance of the T3 medium at the instant when the velocity is v ; hence the accelerative force by which the falling', body is urged at such moment is expressed by 9 u Now a being the space descended by the body in the time t, and v being the velocity as before, an accelerative force is represented de a d v by and by Tire. [Fonce.] Therefore ; whence gdt = dv and integrating this equation, observing that v = 0 when 1 u + fi = 0, we have t = hyp. log. 17-17, 2t ', or hyp. log. u + or 29t again, passing from logarithms to numbers, u + v u = e (c being the 2gt base of the hyperbolic logarithms); whence v = this 1 + e u second member being developed, gives v = gt &c. Substi tilting, in this equation, for r, and again integrating, we have 1 a = (47 These equations for a and v give the space descended and the velocity acquired at the end of any given time t from the moment when the motion commenced. For tables of the values of u (the terminal velo cities) for iron balls, see Dr. If utton's Tracts, tract 37.
Next, let a body be projected vertically upwards in a uniformly resisting medium with an initial velocity = v; and let the body ho of a spherical form so that u may be the same as before : then, the force of gravity and the resistance of the medium acting in a direction opposite to that of the projectile force, we have now g ; (r.
whence gdt This equation, being integrated, gives 1 gt = u arc tan. = + coast. ; and considering that v = v when t = 0, the constant is equal to is arc tan.= putting c to repro - gt sent this term, we have 7 = arc ten. ; and passing from arca to o gt o Ut tangents, we have = ten. u , or v = . Multiplying both members of the last equation by dt, and putting dx for its equal c et rdt, tlrls ovation becomes dx = is tan. dt; which integrated D c gives = hyp. log. cos. u + coast. The constant is determined by considering thatx = 0:when t = 0; whence coast. = hyp. log. cos.
c - gt u u= cos.
therefore x = - hyp.
cos. II Making v = 0 in the above equation for r, we get the value of t when the body has attained its greatest height ; and substituting the value of t so found in the last equation for x, we have that greatest height.
When arrived at the greatest height, the body would begin to return towards the earth ; and it may be shown that the velocity acquired by the body on arriving at the place from whence it was projected would be less than the initial velocity; also that the time of the descent would differ from the time of ascent.
If we imagine the earth to be perforated in the direction of a diameter ; and if a body be allowed to descend towards the centre in a non-resisting medium from any point iu the line of perforation : the law of attraction being in such a case directly proportional to the distance of the body at any time from the centre of the earth (Newton, lib. i. prop. 73), the relation between the space descended and the time of descent may be thus investigated.