FORMULE FOR UNIFORM MOTION.

Here t the time occupied by the movement of a body. g the velocity acquired after a second of time.

=

41

the rapidity of motion acquired after the entire time t.

s = the space passed over by the body in the time t.

By these formulæ it is obvious that any two of the four data being given, the other two are readily found.

CHAPTER IV.

EFFECTS OF GRAVITATION.

Motion produced by Gravity, 55. Action of the Earth on falling Bodies, 56-62. Formula for Gravitation, 57-59. Motion of Projectiles, 61. Bodies falling down inclined Planes, 62. Resistance of Media, 63. Rotation of moving Bodies on their Axes, 64-65. Momentum, 66. Collision of Bodies, 67-71. Bodies falling down Series of Planes, 72. Oscillation in Cycloidal Curves, 73. Pendulum, 74-84. Isochronism of Cycloidal Oscillations, 75. Determination of Force of Gravity by the Pendulum, 78. Formula for, 79-80. Centre of Oscillation, 83. Compensating Pendulum, 84.

55. AMONG the forces which are the most energetic in producing motion on the surface of our globe is the attraction of gravitation (25); this force, whilst acting on bodies under its influence and approaching the earth, is an uniformly accelerating force, becoming as uniformly retarding on bodies receding from the earth; so that a body acted upon by it, passes through different portions of space in different times, and, whilst approaching the earth, would each instant pass through a greater space than that which it traversed in the preceding instant of time. If a ball be let fall from the hand, it can be readily caught during the first few inches of its path, but its velocity afterwards so rapidly increases, that it cannot be intercepted by the most agile arm without difficulty. Even if the descending body fall obliquely, still the same rapid increase of velocity is perceived; this is well illustrated by the falling of bodies down steep descents, or long inclined planes: for the first few yards the mass appears to move slowly, rapidly,

ACTION OF EARTH ON FALLING BODIES.

43

however, it increases in velocity, and, as well illustrated by the fall of a granite block from an alpine ridge of rock, or of the more terrific avalanche acted upon by the increasing intensity of gravitation, it tears its hurried flight through almost any obstacle it encounters.

56. A body left free to move, and acted upon directly by the force of gravitation, all opposing forces being excluded, falls in the latitude of Greenwich at the rate of 16·0954 feet in a second of time, acquiring by this motion a velocity of 32.1908 feet, or 386-2894 inches per second. The space traversed by a falling body in a second, is very nearly equal to 16 feet 1 inch; which is sufficiently correct for ordinary calculations, and to enable us to avoid decimals, which are very inconvenient, unless we use logarithms to lessen the number of figures.

57. When a body sufficiently dense and compact to permit us to disregard the opposition of the medium traversed by it, is acted on by gravitation, it is found that the spaces described by a falling body increase as the squares of the times increase; thus calling 16 feet

=

9, we find that in 1= 3g, in 2d second of time.

3

4

5

6

(3' 9) 4= 5g, in 3d

ditto.

Therefore in 1, 2, 3, 4, 5, &c. seconds, the

spaces traversed

by a falling body are equal to g, × the odd numbers 3, 5, 7, 9, &c. respectively.

Thus, by knowing the time occupied by the falling of any dense body of small bulk, the space traversed by it can be readily calculated; if a bullet falling from a certain height reaches the earth in 3 seconds, we know that

and the space traversed by it is equal to 144 feet 9 inches. This and similar questions can be more readily determined by means of the formulæ for uniformly accelerated motion already given (54),

As an example of this formula, suppose we wish to know the space passed through by a body occupying 23 seconds in its descent, then by logarithms

which is the logarithm of 8514-6 feet, the space traversed in 23" by the body.

58. A still simpler formula may be adopted, and the calculations easier effected, by squaring the time in which the falling body passes through any space, and multiplying this product by the space passed through in a second of time. This formula expressed algebraically, calling the space passed through in a second or 16 ft. = s, is sť2; this, applied to the last question of a body occupying 23 seconds in its descent, is

23 × 23529, and 529 × 16·0954=8514.4666 feet;

or by logarithms,

log. 23 × 23 =2.72346

log. s or 16.0954=1.20670

3.93016 equal to 8514.6 feet.

FORMULE FOR GRAVITATION.

45

In this manner the height of any lofty building or depth of a well or shaft may be determined; for by letting fall a pebble from the top of the one, or into the mouth of the other, and noting the number of seconds which elapse before the sound of its striking the ground or water is heard, then-on squaring this number of seconds, and multiplying the product by 16 feet or, more accurately, by 16.0954 feet, the height of the building or distance of the water from the mouth of the well may be discovered.

59. Also, knowing the time required for the fall of any body through a given space, we can readily discover the velocity with which it moves, and by knowing its velocity we can of course ascertain the time required for its fall through any given space. The following three formula will be sufficient to answer every question connected with this subject; v being the velocity of the falling body, and t the time of its descent, the other letters retaining their former value (54),

60. If a body, instead of being acted upon by gravitation alone, be projected downwards with a given velocity per second; this is to be taken into account, and being expressed in feet, and multiplied by the number of seconds, the product is to be added to the space, also expressed in feet, which the body would have traversed in the same time, if acted upon by the force of gravitation alone. If, on the contrary, the body be projected perpendicularly upwards, its course being opposed to the attraction of gravitation, instead of being added, the effect of the latter is to be subtracted from the space passed through by the projectile, if acted upon by the force of projection only. The following examples will illustrate these statements:

(A.) To what height will a body rise in 3 seconds if projected upwards with a velocity of 100 feet per second?