920
THEORY AND PRACTICE OF ENGINEERING.
Book II.
If an elastic body A impinges a hard and elastic body, C F, at B,it will be so reflected from it that the angle of reflection will beequal to the angle of incidence. 'Die motion at B, parallel to thesurface, not being altered by the stroke, elastic bodies recoveringtheir figure in the same time that they lose it by the stroke;therefore the velocity in the direction of BE is the same after asbefore the stroke, and we have two equal and similar triangles.
But as no two bodies in nature are perfectly elastic, one angle willbe more acute than the other : a non-elastic body striking anotherin a similar state loses half as much motion as if both were per-fectly elastic ; non-elastic bodies only stop, but elastic recede withthe same velocity that they meet with.
On the force of Gravity . — The earth, whose diameter is about 8000 miles, has from itsmagnitude the power of attracting all smaller bodies to its surface; so that when put inmotion they tend, if unopposed in their descent, in a straight line towards its centre : thisis sometimes called the earth’s attraction, or terrestrial gravity, and the force by which thebody is attracted is called weight.
The force of gravity acts equally on bodies in motion and at rest: when put in motionit draws them down, and destroys in some measure the forces which put them into motion,and the same quantity of force is required to keep them in uniform motion directlyupwards, suspended, or at rest; when they descend uniformly that force which is suf-ficient to prevent their acceleration in descending is equal to their weight.
When a heavy body falls from any height, as it approaches the ground it increases invelocity ; this is rendered evident from the force with which it strikes the ground, thatforce increasing in proportion to the height from whence it has fallen, and also to itsvelocity at the moment it reaches the ground. The velocities of falling bodies are theaccumulated effects of the attraction of the earth during the entire time of their descent:at each instant a new impulse is given which increases the velocity, so that the finalvelocity is the sum of all which have been communicated. Gravity is, then, a uniformforce or attraction, which affects all bodies equally, whatever may be their position. Thevelocity of a falling body is as the time of its falling from a state of rest, the force of gravitybeing the quantity of matter.
All bodies falling by their own weight gain equal velocities in equal times, and whenthe same bodies are thrown or propelled upwards, they lose as much in an equal time :when a body falls through a certain space in a given time, it will afterwards descendthrough four times that space in the next similar period of time; through nine times thatspace in the third, sixteen times in the fourth, &c.: or, to ascertain the space through whicha body will fall in a given number of seconds, multiply the space in the first second oftime by the squares of the total number of seconds it was in falling; thus the spaces throughwhich a body falls is as the square of the time from the commencement to the fall, or thevelocity is as the square root of the height fallen, and it is absolutely necessary, before wecan compute any velocity, that we should know how much it would fall through in asecond of time. The velocity acquired in a second, and the space fallen through in asecond, are the elements upon which the whole calculations are formed: when one ofthese elements are known, the other may be immediately found, as there is a remarkablerelation between the space through which a body falls, and the velocity it acquires in anygiven time. Til the latitude of London a body will fall in a single second 193 inches, orabout 16 feet. Taking any equal parts of time, the spaces described by a falling body ineach successive part of time will be the odd numbers 1, 3, 5, 7, 9, 11, &c.
The spaces, velocities, and times may be judged of by the following table: the spacefallen through in the first second of time is considered the unit of length: —
h'tg. 122$.
Seconds from tbebeginning of theDescent.
Velocity acquiredat the end ofthat Time .
Space fallenthrough.
Space fallenthrough in the lastsecond of the Fall.
1
2
1
1
2
4
4
3
■ 3
6
9
5
4
8
16
7
5
10
25
9
6
Id
36
11
7
14
49
13
8
16
64
15
Between the three quantities, the height, the time, and the final velocity, there are fixedrelations. The time counted from the commencement of the fall and the final velocity are