94. Formal statement of the laws of falling bodies. Putting together the results of the last two sections, we obtain the following table, in which D represents the distance traversed during the first second in any uniformly accelerated motion.

Since D was shown in § 93 to be equal to half the acceleration a, we have at once, by substituting

line of the table,

v = at;
S = at2.

a for D in the last

(1)

(2)

These formulas are simply the algebraic statement of the facts brought out by our experiments, but the reasons for these facts may be seen as follows:

Since in uniformly accelerated motion the acceleration a is the velocity in centimeters or feet per second gained each second, it follows at once that when a body starts from rest, the velocity which it has at the end of t seconds is given by vat. This is formula (1).

To obtain formula (2) we have only to reflect that distance traversed is always equal to the average velocity multiplied by the time. When the initial velocity is zero, as in this case, and the final velocity is at, average velocity (0+ at) ÷ 2 = at. Hence S = at 2.

This is formula (2).

These are the fundamental formulas of uniformly accelerated motion, but it is sometimes convenient to obtain the final velocity v directly from the total distance of fall S, or vice versa. This may be done, of course, by simply substituting in (2) the value of t obtained from (1); namely, v/a. This gives

The most significant and far-reaching of the advances of the twentieth century, - man's conquest of the air after centuries of failure was made when the Wright brothers first introduced the principle upon which all successful flight by heavier-than-air machines now depends, namely, control of stability by the warping of wings, or by ailerons (hinged attachments to wings), in connection with the use of a rudder. The upper panel shows one of the original gliders (Wilbur Wright inside) with which the Wrights first mastered the art of gliding (19001903) and made more than a thousand gliding flights, some of them 600 feet long, following in this work the principles of gliding flight first demonstrated by Lilienthal and a little later, much more completely, by Chanute of Chicago (1895-1897). The lower panel shows "the first successful power flight in the history of the world" (Orville Wright in the machine, Wilbur running beside it as it rose from the track). Four such flights were made on the morning of December 17, 1903, the longest of which lasted 59 seconds and covered a distance of 852 feet against a 20-mile wind

The Wright brothers, the men who first solved the age-old problem of mechanical flight, were born
in Dayton, Ohio. They received a high-school education, had much mechanical ability, and were
flight enthusiasts from early youth, devoting their time almost exclusively to this problem. They
were thirty-six and thirty-two years old, respectively, when they made the historic flight at Kitty-
hawk, N.C., in 1903. Two years later at Dayton they made the first successful long-distance flight

Velocities

V =

V = 32.16

Distances

in feet

0

= S

(16.08)

16.08 = S

95. Acceleration of a freely falling body. If in the experiment discussed in §93 the slope of the plane is made steeper, the results will obviously be precisely the same, except that the acceleration has in ft. per sec. a larger value. If the board is tilted until it becomes vertical, the body becomes a freely falling body (Fig. 78). In this case the distance traversed the first second is found to be 490 centimeters, or 16.08 feet. Hence the acceleration is 980 cm. per second each second, or 32.16 ft. per second each second. This acceleration of free fall, called the acceleration of gravity, is usually denoted by the letter g. For freely falling bodies, then, the three formulas of the preceding paragraph become

v = gt,

S = 1 gt2,

v = √2 gs.

(48.24)

V = 64.32 64.32 = S

(80.40)

To illustrate the use of these formulas, suppose we wish to know with what velocity a body will hit the earth if it falls from a height of 200 m., o 20,000 cm. From (6) we get

r = √2 × 980 × 20,000

= 6261 cm. per second.

The accurate determination of g is never made by direct measurement, for the laws of the pendulum established in § 83 make this instrument by far the most accurate one obtainable for this determination. It is necessary only to measure the length of a long pendulum and the time t between two successive passages of the bob across the mid-point, and then to substitute in

the formula t = π √g in order to obtain g with a high degree of precision. The deduction of this formula is not suitable for an elementary text, but the formula itself may well be used for checking the value of g.

96. Height of ascent. If we wish to find the height S to which a body projected vertically upward will rise, we reflect that the time of ascent must be the initial velocity divided by the upward velocity which the body loses per second, that is, t=y/g; and the height reached must be this multiplied by the average velocity v + 0 , that is,

2

S

=

v2 2 g

or v = √2 gs.

(7)

Since (7) is the same as (6), we learn that in a vacuum the speed with which a body must be projected upward to rise to a given. height is the same as the speed which it acquires in falling from the same height.

97. Path of a projectile. Imagine a projectile to be shot along the line ab (Fig. 79). If it were not for gravity and the resistance

of the air, the projectile would travel with uniform velocity along the line ab, arriving at the points 1, 2, 3, etc. at the end of the successive seconds. Because of gravity, however, the projectile would be vertically below these points by the distances 16.08 ft., 64.32 ft., 144.72 ft., etc. Hence it would follow the path indicated by the dotted curve (a parabola). But because of air resistance the height of flight and the range are diminished, and the general shape of the trajectory is similar to the continuous curved line. Fig. 80 represents bomb-dropping by an airplane.