Page:Encyclopædia Britannica, Ninth Edition, v. 11.djvu/78

Rh already referred to is descending with an uniform velocity of 32 2 feet per second. A body let drop from the ceiling during the motion, will, as before, reach the floor in one second. The body will therefore have acquired, relatively to the moving lift, a velocity of 32 &quot;2 feet per second. But the lift is itself in motion with a velocity of 32 2 feet per second. The actual velocity of the body must be measured by its velocity relatively to the lift, added to the velocity of the lift itself. It therefore appears that the body which, when it commenced to fall, had a velocity of 32 2 feet per second, acquires an equal amount during its fall, so that at its close the body actually had a velocity of 32 2 + 32 2 = 64 4 feet per second. A body falling freely from rest acquires a velocity of 32 2 feet in the first second; it follows that at the close of the first second the body is in the same condition as if it were let fall from the ceiling of the lift, under the circumstances just described. The motion during the third second is therefore commenced with the velocity of 64 4 feet, and in consequence of this initial velocity alone a distance of 64 - 4 feet will be accomplished in the third second. To this must be added IG l feet, being the additional distance due to the action of gravity, and there fore we have for the distance through which a body falling freely from rest will move in the third second, 6 4 4 + IG l = 80 - 5. Similar reasoning will show that the velocity ac quired at the close of the third second is G4 4 + 32 2 = 9G G. With this velocity the fourth second is commenced, and therefore the distance accomplished during the fourth second is 96-6 + 16-1 = 1127. The results at which we have arrived may be summarily stated in the following propositions : A body falling freely from rest acquires a velocity which is equal to the product of 32 2 and the number of seconds during which the motion has lasted. A body falling freely from rest moves over spaces propor tional to the consecutive odd mimbers (1, 3, 5, 7, &c.) in each of the consecutive seconds during which the motion lasts. A body falling freely from rest will, in a given number of seconds, move over a distance ivhich is found by multiplying the square of the number of seconds by IG l.

§&thinsp;3. Values of g.&mdash;The velocity acquired by a body in one second is usually denoted by the symbol g. The following are values of g at different parts of the earth (adapted from Everett On C.G.S. Units, p. 12): Latitude. Value of g in Feet per Second. Length in Feet of Pendulum beating Seconds. Equator 32-091 S 2514 Latitude 45 45 32-173 3-2597 Munich 48 9 32-181 3-2607 Paris 48 50 32-183 3-2609 Greenwich 51 29 32-191 3-2616 Gottingen 51 32 32-191 3 2616 Berlin 52 30 32-194 3-2619 Dublin 53 21 32-196 3-2621 Manchester 53 29 32-196 3-2622 Belfast 54 36 32-199 3 2625 Edinburgh 55 57 32-203 3-2629 Aberdeen 57 9 32-206 3-2632 Pole. 90 32 255 3-2682 The value of g in feet at a station of which the latitude is A, and winch is h feet above the level of the sea, is in feet g = 32 -173 -0-082 cos 2A-0-000003&. The length of the pendulum in feet which vibrates in one second is 1 = 3 -2597 -0-0083 cos 2A-0-OOOOOOA. g is really the excess of gravitation over the centrifugal force arising from the earth s rotation. The value of gravitation alone is given by the following expression : 32-225-0-026 cos 2A.

§&thinsp;4. Algebraical Formulæ.&mdash;The employment of the symbols and operations of algebra will enable us to express very concisely the results at which we have arrived. Let v denote the velocity acquired in t seconds by a body -which has been dropped from a state of rest. Let s denote the number of feet over which the body has moved. The laws we have arrived at may be thus expressed : v = gt ; s = 5gP. From these equations we can eliminate t and obtain v- = 2gs. This expresses the velocity acquired in terms of the distance through which the force has acted. We have hitherto considered the movement of a falling body which was simply dropped. It remains to determine the elfect on the movement of the body which would be produced by a certain initial velocity. Let us for simplicity take the case of a body thrown vertically downwards, and calculate the distance through which the body will move in a certain time, as well as the velocity which it will acquire. In the act of throwing the hand moves with a certain velocity, and the body when released starts off with that velocity. It will thus be observed that the act of throwing is merely to impart initial velocity to the body. Let v 1 be the initial velocity with which the body leaves the hand. Then the velocity of the body at the moment of starting is precisely the same as it would have been had it been dropped from rest v -r-g seconds previ ously. The velocity acquired at the end of t seconds is therefore the same as would have been acquired by a body which fell from rest for a period of (v -^-g + t) seconds ; whence we have = v + gt. The distance must obviously be equal to the difference between the distance through which the body would drop from rest in (v -^-g + t} seconds and the distance through which a body would drop from rest in v -i-g seconds ; whence The case of a body projected vertically iipwards seems at first to present somewhat greater difficulties, but this is not really the case. Such problems can always be readily solved by the help of the following general principle : A body moving vertically for t seconds will, at the end of that time, be 5yt 2 feet nearer the earth tkanit would have been had gravity not acted. If the body be projected vertically upwards with an initial velocity v, then, if the influence of gravity were suspended, the body would in t seconds ascend to a height v t in accordance with the first law of motion. The effect of gravity will be to reduce the height actually obtained by the amount -5yt 2 . Whence wo have s v t- 5gt 2 This expression may be written in the form s=v *+2g- -5g(t -v +g) It is therefore obvious that the greatest altitude h is attained when t = v +g ; in which case h = v ^2g, or v * = 2gh. As an illustration we may take the case of a body thrown verti cally upwards with an initial velocity of 40 feet per second, and in quire where that body will be at the end of two seconds. Had gravity not acted, the body would, in two seconds, have ascended to a height 2 x 40 = 80 feet. The action of gravity will reduce this by &quot;5&amp;lt;/2 2 = 2&amp;lt;7 feet, and hence the actual height of the body will be 80 - 2g feet, = 1 5 6 if g be taken at 32 2.

§&thinsp;5. Motion of a Projectile.&mdash;We have hitherto referred only to the motion of a falling body in a vertical line ; it will now be neces sary to examine some cases in which the motion of the body is not so restricted. From a point on the mast of a steamer g-^-2 feet above the deck a ball is dropped, which falls upon the deck at a certain point. When the steamer is at rest the time taken by the ball to fall will of course be one second, and its path will be vertical ; but when the steamer is moving with uniform velocity it is found that the ball still falls precisely on the same spot of the deck as when the steamer was at rest, and that the time occupied in the descent is still one second. It is obvious that in this case the ball does not move in a straight line at all, but in a curved path due to the motion of the vessel compounded with the actual falling motion. We there fore see that the effect of the motion of the vessel on the ball was to project that ball with a certain initinl horizontal velocity, but that notwithstanding that initial velocity the ball still reaches the deck in one second. We are therefore led to the general conclusion that A body projected horizontally will, at the end of t seconds, hare fallen through a space of 16 1 t 2 feet. An experiment illustrating this result maybe made in an exceedingly simple manner. Take a marble in each hand, and throw one of the marbles horizontally at the same time as you drop the other