(2.) Suppose light to move at the rate of 192,000 miles in a second of
time, how long will it take to reach the earth from the sun, which is 95
millions of miles distant 1 Ans. 8 tnin. 14.07 we. +
(3.) If a railroad-car run at the rate of 20 miles an hour, how long will
it take to go from Washington to Boston, distance 432 miles ? Ana. 21.6 hr.
(4.) Suppose a ship sail at the rate of 6 miles an hour, how long will it
take to go from the United States to Europe, across the Atlantic Ocean, a
distance of 2800 miles 1 Ans. 19 da. 10 hr. 40 min.
(5.) [f the earth go round the sun in 365 days, and the distance travelled
be 540 millions of miles, how fast does it travel 1 Ans. 1,479,452 s 4 w mi.
(6.) Suppose a carrier-pigeon, let loose at 6 o'clock in the morning from
Washington, reach New Orleans at 6 o'clock at night, a distance of 1200
miles, how fast does it fly 1 Aw. 100 mi. per hr.
How may the
*rL,Sy 148 - The P P as8ed over ma y bo foi]nd fe y
in motion be multiplying the velocity by the time.
Miles per hour. Feet } ft second
Slow rivers 3 ... 4
Rapid rivers 7 10
Moderate wind 7 1C)
A storm . 36 52
A hurricane 80 117
f'.rnmon musket-ball ... 850 1,240
Rifle-ball 1,000 1,466
'24 lb. cannon-ball .... 1,600 2,346
Air rushing into a vacuum )
dt 32^ F 5 W ' ' ' 1 2%
\ir-gun bullet, air com- S
pressed to '01 of its V 466 . . . 683
Sound , 743 ..... . 1,142
A point on the surface of > , n 7 , , '
the earth \ 1)037 1 ' 520
Earth in its orbit .... G7,374 98,815.
The velocity of light is 186,000 miles in a second of time.
The veJo:ity of the electric fluid is said to be still greate dad sow*
luthonciei .sti-te it to be at tho rate of 288 000 miles in a second *x time.
44 NATURAL PHILOSOPHY
Thus, if the velocity be 5 miles an hour, and the time 20 hours
she space will oe twenty multiplied by 5. Ans. 100 miles.
144. (1.) If a vessel sail 125 miles in a day for ten days, how far will it
&iil in that time '? Ans. 1250 mi.
(2.) Suppose the average rate of steamers between New York and Aloan?
be about 11 miles an hour, which they traverse in about 14 hours, what
is the distance between these two cities by the river 1 Ans. 154 mi.
(3.) Suppose the cars going over the railroad between these two citie?
travel at the rate of 25 miles an hour and take 8 hours to go over the dis-
tance, how far is it from New York to Albany by railroad 1 Ans. 200 mi,
(4.) If a man walking from Boston at the rate of 2 miles in an hour reach
Salem in 6 hours, what is the distance from Boston to Salem 1 Ans. 15 mi.
(5.) The waters of a certain river, moving at the rate of 4 feet in a
second, reach the sea in 6 days froin the time of starting from the source
of the river. What is the length of that river 1 Ans. 392-jiy mi.
(6.) A cannon-ball, moving at the rate cf 2400 feet in a second of time,
strikes a target in 4 seconds. What is the distance of the target! A. 9600 ft
145. The following formulae embrace the several ratios of the time, space
and velocity :
(1.) The space divided by the time equals the velocity, or = v>
(2.) The space divided by the velocity equals the time, r - = t.
(3.) The velocity multiplied by the time equals the space,
How many _ 146. There are three kinds of Motion
are there ? namely, Uniform, Accelerated and Retarded.
What is 147. When a body moves over equal spaces in
Uniform , .. ., '. . . , , _ _ .
Motion? equal times, the motion is said to be uniform.
What is 1^' Wh en ^ ne spaces or distances over which
Accelerated a body moves in equal times are successively
greater, the motion is said to be Accelerated.
What is -^. Wh en the spaces for equal times are
Retarded successively less, the motion is called Retarded
Motion? ,.. ,.
How are Uni- 150. Uniform Motion is produced by the
form,Acceler- ,. ,
ated and jRe- momentary action of a single force. Accel-
tarded Motion era t e d Motion is produced by the continued
produced ? action of one or more forces. Retarded Mo-
tion is produced by some resistance.
151. A ball struck by a bat, or a stone thrown from the hand, is
in theory an instance of uniform motion ; and if the attraction of
gravity and the resistance of the air could be suspended, it would
proceed onward in a straight line, with a uniform motion, forever.
But as the resistance of the air and gravity both tend to deflect it,
it in fact becomes first an instance of retarded, and then of accel-
152. A stone, or any other body, falling from a height, is an
instance of accelerated motion. The force of gravity continues to
operate upon it during the whole time of its descent, and con-
stantly increases its velocity. . It begins its descent with the first
impulse of attraction, and, could the force of gravity which gave it
the impulse be suspended, it would continue its descent with a
uniform velocity. But, while falling it is every moment receiving a
new impulse from gravity, and its velocity is constantly increasing
during the whole time of its descent.
153. A stone thrown perpendicularly upward is an instance of
retarded motion ; for, as soon as it begins to ascend, gravity immedi-
ately attracts it downwards, and thus its velocity is diminished. The
retarding force of gravity acts upon it during every moment of its
ascent, decreasing its velocity until its upward motion is entirely
destroyed. It then begins to fall with a motion continually acceler-
ated until it reaches the ground.
does a body ^54. A body projected upwards will occupy the
occupy in its . .. , ,
ascent and same time m lts ascent an( l descent.
This is a necessary consequence of the effect of gravity, which
uniformly retards it in the ascent and accelerates it in its descent.
155 ; P*PKTnAL MOTION. - Perpetual Mo-
be produced? tion is deemed an impossibility in mechanics,
because action and reaction are always equal and in con-
156 ' B J the aCti H f a bod 7 is meant thc
Reaction ? effect which it produces upon another body.
By reaction is meant the effect which it receives from the
body on which it acts.
Thus, when a body in motion strikes another body, it acts upon it
or produces motion ; but it also meets with resistance from the body
whicb ip struck, and this resistance is the reaction of the body.
46 NATURAL PHILOSOPHY.
Uliistratioji of Action and Reaction by tieaiis of Elastic and
(1.) Figure 6 represents two ivory * balls, A and B,
of equal size, weight, &c., suspended by threads. If the
ball A be drawn a little on one side and then let go,
it will strike against the other ball B. and drive it off A
to a distance equal to that through which the first ball
fell ; but the motion of A will be stopped, because when it strikes
B it receives in return a blow equal to that which it gave, but in
a contrary direction, and its motion is thereby stopped, or, rather,
given to B. Therefore, when a body strikes against another,
the quantity of motion communicated to the second body is lost
by the first ; but this los proceeds, not from the blow given by
the striking body, but from the reaction of the body which it
(2.) Fig. 7 represents six ivory balls of equal weight, suspended
by threads. If the ball A be drawn out, of the perpendicular
and let fall against B. it will communicate its mo-
tion to B, and receive a reaction from it which will
stop its own motion. But the ball B cannot move
without moving ; it will therefore communicate
the motion which it received from A to C. and
receive from C a reaction, which will stop its motion.
In like manner the motion and reaction are received by each ol
the balls D, E, F ; but, as there is n ball beyond F to act upotj
it, F Will fly off.
N. B. Thi experiment is to bt performed -vith elastic balls . i ly.
(3). Fig. 8 represents two tails of clay (which are r ot elastic*
of equal weight, suspended by s* rings. If the ball D
be raised and let fall against E, oTily part of the mo- Fg. 8.
tion of 1) will be destroyed by it (because the Dodies
ai > non- elastic), and the two balls will move on togeth-
er to and e, which are less distant from the ver-
tical line than the ball D was before H foil. Still,
* It will be recollected that ivory is considered highly elastic.
however, action and reaction are equal, for the action on E ia
only enough to make it move through a smaller space, but sc
much of D's motion is now also destroyed.
157. It is upon the principle of action and reaction that birds
are enabled to fly. They strike the air with their wings, and the
reaction of the air enables them to rise, fall, or remain stationary,
at will, by increasing or diminishing the force of the stroke of their
158. It is likewise upon the same principle of action and reaction
that fishes swim, or, rather, make their way through the water,
namely, by striking the water with their fins. J
159. Boats are also propelled by oars on the same principle, and
the oars are lifted out of the water, after every stroke, so as com
pletely to prevent any reaction in a backward direction.
^How may 160. Motion may be caused either by action ox
motion be . ^ TT1 i , .. ... -MI
caused? reaction. When caused by action it is callecf
Incident, and when caused by reaction it is called Reflected
* Figs. 6 and 7, as has been explained, show the effect of action ancr re-
action in elastic bodies, and Fig. 8 shows the same effect in non-elastic bodies.
When the elasticity of a body is imperfect, an intermediate effect will bo
produced ; that is, the ball which is struck will rise higher than in case of
non-elastic bodies, and less so than in that of perfectly elastic bodies; and
the striking ball will be retarded more than in the former case, but not
stopped completely, as in the latter. They will, therefore, both move
onwards after the blow, but not together, or to -the same distance ; but in
this, as in the preceding cases, the whole quantity of motion destroyed in
the striking ball will be equal to that produced in the ball struck. Con-
nected with " the philosophical apparatus " is a stand with ivory balls, to
give a visible illustration of the effects of collision.
f The muscular power of birds is much greater in proportion to their
wei-ght than that of man. If a man were furnished with wings sufficiently
large to epable him to fly, he would not have sufficient strength or muscular
power to put them in motion.
J.The power possessed by fishes, ot sinking or rising in the water, ia
greatly assisted by a peculiar apparatus furnished them by nature, called
aii air-bladder, by the expansion or contraction of which they rise or fall,
an the principle of specific gravity.
The word incident implies falling upon or directed towards. The word
reflected implies turned ~back. Incident motion is motion directed towards
any particular object against which a moving body strikes. Reflected mo-
tion is that which is caused by the reaction of the body which is struck.
Thus, when a ball is thrown against a surface, it rebounds or is turned
back. This return of the ball is called reflected motion. As reflected mo-
tion is caused by reaction, and reaction is increased by elasticity, it follows
that reflected motion is always greatest in those bodies which are most elas-
tic. For this reason, a ball filled with air rebounds better than one stuffed
with bran or wool, because its elasticity is greater. For the same reason,
balls made of caoutchouc, or India-rubber, will rebound more than those
which are made of most other substances.
What, u 161. The angle * of incidence is the angle formed
of a jlna- by the line which the incident body makes in its
detue? passage towards any object, with a line perpendic-
ular to the surface of the object.
* As this book may fall into the hands of some who are unacquainted with
geometrical figures, a few explanations are here subjoined :
1. An angle is the opening made by two lines which meet each other in a
point. The size of the angle depends upon the opening, and not upon tht length
of the lines.
2. A circle is a perfectly round figure, every
part of the outer edge of which, called the cir-
cumference, is equally distant from a point
within, called the centre. [See Fig. 9.]
3. The straight lines drawn from the centre
to the circumference are called radii. [The
singular number of this word is radius.} Thus,
in Fig. , the lines CD, C 0, C R, and C A, are
4. T^e lines drawn through the centre, and
terminating in both ends at the circumference,
are called diameters. Thus, in the same figure, D A is a diameter of th
5. The circumference of all circles is divided into 360 equal parts, called
degrees. The diameter of a circle divides the circumference into two equal
parts, of 180 degrees each.
6. All angles are measured by the number of degrees which they contain
Thus, in Fig. 9, the angle R C A, as it includes one-quarter of the circle, is
an angle of 90 degrees, which is a quarter of 360. And the angles R C
and C D are angles of 45 degrees.
7. Angles of 90 degrees are right angles ; angles of less than 90 degrees,
acute angles; and angles of more than 90' degrees are called obtuse angles,
Thus, in Fig. 9, RC A is a right angle, C R an acute, and C A an obtuse
8. A perpendicular line is a line which makes an angle of 90 degrees on
each side of any other line or surface ; therefore, it will incline neither to
the one side nor to the other. Thus, in Fig. 9, R C is perpendicular to D A.
9. The tangent of a circle is a line which touches the circumference, with-
out cutting it when lengthened at either end. Thus, in Fig. 9, the line RT
Is a tangent.
10. A square is a figure having four equal sides, and four equal angles.
These will always be right angles. [See Fig. 11.]
11. A parallelogram is a figure whose opposite sides are equal and parallel
[See Figs. 12 and 13.] A square is also a parallelogram.
12 A rectangle is a parallelogram whose angles are right angles.
[N. B. It will be seen by these definitions that both a square and u
rectangle are parallelograms, but all parallelograms are not rectangles nor
equares. A square is both a parallelogram and a rectangle. Three thing*
ure essential to a square; namely, the four sides must all be equal, they must
ttlso be parallel, and the angles must all be right angles. Two things only
lire essential lo a rectangle ; namely, the angles must all be right angles,
and the opposite sides must be equal and parallel. One thing only is essen-
tial to a parallelogram; namely, the opposite sides must be equal and
13 The diagonal of & square, of a parallelogram, or a rectangle, \a e liu
Explmn 162. Thus, in Fig. 10, the line Fig. 10.
Fig. 10 ABC represents a wall, and P B -^
a line perpendicular to its surface. O is a _ """".
ball moving in the direction of the dotted ^. - **
line, B. The angle O B P is the angle of R - ""
What is 163. The angle or reflection is the angle formed
tfr^fiec- ty ^ e P er P en dicular with the line made by the
iion ? reflected body as it leaves the surface against
which it struck.
Thus, in Fig. 10, the angle P t R is the angle of reflection.
164. The angles of incidence and re-
of incidence to the flection are always equal to one another.*
angle of reflection ?
(1.) Thus, in Fig. 10, the angle of incidence, B P, and the
angle of reflection, P B R, are equal to one another ; that is,
they contain an equal number of degrees.
What will be the 165 From what has now been gtate a w jth
course of a body . . in
in motion which regard to the angles of incidence and renec-
strikes against t it f u owg tna t when a ball is thrown
unothen fixed .
)ody ? perpendicularly against an object whick
it cannot penetrate, it will return in the same direction ,
but, if it be thrown obliquely, it will return obliquely on
f ,he opposite side of the perpendicular. The more 06-
liquely the ball is thrown, the more obliquely it will
drawn through either of them, and terminating at the opposite angles. Thus,
In Figs. 11, 12, and 13, the line A C is the diagonal of the square, parallelo-
gram, or rectangle.
* An understanding of this law of reflected motion is very import? nt,
because it is a fundamental law, not only in Mechanics, but also in Pyro-
:iomics, Acoustics and Optics.
t It is from a knowledge of these facts that skill is acquired in many
different sorts of games, as Billiards, Bagatelle, <fec. A ball may also, on
tho aaine principle, be thrown from a gun against a foi tificatiu'u so a* t*
roach a,u object out of the range of a direct shot.
50 NATURAL PHILOSOPHY.
What is the 166 ' MOMENTUM. The Momentum* of a
Momentum of body is its quantity of motion, f and implies
an expression of weight and velocity at the
How is the The Momentum of a body is ascertained
Momentum of a , ,.. , . .. ..,,,.,
lody calculated? D J multiplying its weignt by its velocity.
167. Thus, if-the velocity of a body be represented by 5 and iU
eight by 6, its momentum will be 30
How can a 1QS. A small or a light body may be made
ight body to strike against another body with a greater
be mule to force than a heavier body simply by giving it
do as much ~ . . .. , ' . . ,. .
damage as sufficien velocity, that is, by making it hav
a large one ' greater momentum.
Thus, a cork weighing \ of an ounce, shot from a pistol with the
velocity of 100 feet in a second, will do more damage than a leaden
shot weighing of an ounce, thrown from the hand with a velocity
of 40 feet in a second, because the momentum of the cork will b<b
The momentum of the cork is 4 X 100 = 25.
That of the leaden shot is j X 40 =5
169. Questions for Solution.
(1.) What is the momentum of a body weighing 5 pounds, moving with
ch3 velocity of 50 feet in a second 1 Ans. 250.
(2.) ^'hat is the momentum of a steam-engine, weighing 3 tons, moving
with the velocity of 60 miles in an hour ] Ans. 180.
[N. B. It must be recollected that, in comparing the momenta of bodies
the velocities and the time of the bodies compared must be respectively of
the same denomination. If the time of one be minutes and of the other be
t -jura, they must both be considered in minutes, or both in hours. So.
with regard to the spaces and the weights, if one be feet all must b
expressed in feet ; if one be in pounds, all must be in pounds. It is better,
however, to express the weight, velocities and spaces, by abstract numbers
as follows :]
(3.) If a body whose weight is expressed by 9 and velocity by 6 is in
motion, what is its momentum 1 An*, 54.
(4.) A body whose momentum is 63 has a velocUy of 9 ; what is its weight f
* The plural of this word is momenta.
f The quantity of motion communicated to a body does not affect thf
duration of the motion. If but little motion be communicated, the body
rill move slowly. If a great degree be imparted, it will move rapidly.
But in both cases the motion will continue until it is destroyed by some
N 3. Tbe momentum being the product of the weight and velocity, th
weight is found by dividing the momentum by the velocity, and the velocity
J found by dividing the momentum by the weight.]
(5.) The momentum is expressed by 12, the weight by 2 ; what is th
velocity 1 Ans. 6.
(6.) The momentum 9, velocity 9, what is the weight 7 Ans 1.
(7.) -Momentum 36, weight 6, required the velocity. Ans. 6.
(8.) A body with a momentum of 12 strikes another with a momentum of
ti ; what will be the consequence 7 Ans. Both have mom. of 6.
[N. B. When two bodies, in opposite directions, come into collision , they eac*
lo^e an equal quantity of their momenta.}
(9.) A body weighing 15, with a velocity of 12, meets another coming in
the opposite direction, with a velocity of 20, and a weight of 10 ; what wilj
he the effect 7 Ans. Both move *ith mom. of 20.
(10.) Two bodies meet together in opposite directions A has a velocity
of 12 and a weight of 7, B has a momentum expressed by 84. What, wib
^e the consequence 7 Ans. Both mom. destroyed.
(11.) Suppose the weight of a comet be represented by 1 and it's velocity
oy 12, and the weight of the earth be expresse'd by 100 and its velocity bj
10, what would be the consequence of a collision, supposing them to b
tioving in opposite directions 7 Ans. Both have inom. of 988.
(12.) If a body with a weight of 75 and a velocity of 4 run against a mai
,tose weight is 150, and who is standing still, what will be the cons
quence, if the man uses no effort but his weight? Ans. Man has vel. of !
(13.) With what velocity must a 64 pound cannon-ball fly to be equally
effective with a battering-ram of 12,000 pounds propelled with a velocity
of 16 feet in a second 7 Ans. 8000.A
170. ATTRACTION LAW OF FALLING BODIES. When one bod)
strikes another it will cause an effect proportional to its own weight
and velocity (or, in other words, its momentum) ; and the bod?
which receives the blow will move on with a uniform velocity (if
the blow be sufficient to overcome its inertia) in the direction of
the motion of the blow. But, when a body moves by the force of
a constant attraction, it will move with a constantly accelerated
171. This is especially the case with falling bodies. The earth
attracts them with a force sufficient to bring them down through &
certain number of feet dining the first second of time. While the
body is thus in motion with a velocity, say of sixteen feet, the earth
still attracts it, aad during the second second it communicates an
additional velocity, and every successive second of time the attrac-
tion of the earth adds to the velocity in a similar proportion, so that
dniing any given time, a falling body will acquire a velocity which,
in the same time, would carry it over twice the space through which
it has already fallen. Hence we deduce the following law,
\\'J,at ^ the 172. A body falling from a height will fall
ing bodwt sixteen feet in the first second of time,"* three
* This is only an approximation to the truth ; it actually falls
fcnt and one inch during the first second, three times that distance in th
fecoud. &o The questions proposed to be solved assume sixteen feet onlv
52 NATURAL PHILOSOPHY.
times that distance in the second, five times in the
third, seven in the fourth, its velocity increasing during
every successive second, as the odd numbers 1, 3, 5, 7, 9
H, 13, &c.*
The laws of falling bodies are clearly demonstrated by a mechanical
arrangement known by the name of " Attwood^s Machine," in which a small
weight is made to communicate motion to two others attached to a cord
passing over friction-rollers (causing one to ascend and the other to
descend), and marking the progress of the descending weight by the oscil-
lations of a pendulum on a graduated scale, attached to one of the columns
of the machine. It has not been deemed expedient to present a cut of the
machine, because without the machine itself the explanation of its opera-
tion would be unsatisfactory, with the machine itself in view the sim-
plicity of its construction would render an explanation unnecessary.
* The entire spaces through which a body will have fallen in any given
number of seconds increase as the squares of the times. This law was dis-
covered by Galileo, and may thus be explained. If a body fall sixteen feet
in one second, in two seconds it will have fallen four times as far, in three
seconds nine times as far, in four seconds sixteen times as far, in the fifth
second twenty-five times, <fcc., in the sixth thirty-six times, Ac.
ANALYSIS OF THE MOTION OP A FALLING BODY.
Number of Seconds. Spaces. Velocities. Total Space
4 7 8 16
6 9 10 25
6 11 12 36
7 13 14 49
8 15 16 64
9 17 18 81
10 19 20 . - 100
From this statement it appears that the spaces passed through by
falling body, in any number of seconds, increase as the odd numbers 1, 3,
6, 7, 9, 11, &c. ; the velocity increases as the even numbers 2, 4, 6, 8, 10,
12, &c. ; and the total spaces passed through in any given number of
seconds increase as the squares of the numbers indicating the seconds,
thus, 1, 4, 9, 16, 25, 36, &c.
Aristotle maintained that the velocity of any falling body is in direct
proportion to its weight; and that, if two bodies of unequal weight were let
fall from any height at the same moment, the heavier body would reach the