South Bend news-times. (South Bend, Ind.) 1913-1938, 1918-11-29

To illustrate what this means
suppose that. TO begin with. the
body is 5000 miles from the earth's
center, or 1000 miles above its sur
face. and that IT approaches under
the pull of gravitation until it
reaches the surface. when it will
be only 1000 miles from the CEN
ter 1t is now only half as far from
the center of force as il was at first.
and in accordance with the law of
inverse squares, the force of gravita-
ton. will have increased four-fold.
four being the inverse square of -.
or the fraction IA turned upside
down
When the falling body is near the
earth's surface. the force may be re-
edrded as approximately constant,
and the accelerat ton that il produces
is then about 1922 feet per second
| (per second denoted in all mathe-
NAUTICAL formulate by the letter "g").
But as soon as the distance amounts
TO a considerable fraction OF the
earth's radius, the variation of the
force becomes important, and the
problem can then be accurately
solved only by the methods OF the
calculus. which il is impossible to
demonstrate here. But the results
can be indicated. and a formula can
be given. derived. to be sure, from
the calculus, but capable of being
applied by anybody having the
slightest mathematical knowledge.
| Before ating this formula and
giving an example of its applica-
ton, I must remind the reader that
! the law Of bodies, falling under the
force of gravitatiOn, is reversible IN
this sense, viz. that the same velOc-
i'y that a body would acquire in fall
ing any given distance toward the
earth. would suffice TO carry the
body up again to its starting point,
against the force of gravitation
Thus. If a gun can send a bullet
to vertical height of II ga miles,
with an initial velocity of 1000 feet
per second. then in failing back
again from the height the bullet will
acquire By the time il Strikes the
earth the same velocity with which
/ It started to rise. Gin this calculation
\ the air resistance, and the slight
variation in the force Of gravitation
! have been neglected..
| A similar calculation can be made
/ for a ball thrown vertically upward
/ by hand. If your muscles could i'm
/ part TO the ball an initial velocity
4 Of SO feet per second it would rise
1 TO a height of 144 feet, and on fall
/ ing back strike the ground with a
velocity Of 90 feet per second, air
resistance. as before, neglected.
These examples of the effects of
gravitation near the earth serve to
show the relation Of distance TO the
other elements of the problem. Now.
the formula derived from the cal
culuS to which 1 have referred, and,
which serves to tell us. by an easy
calculation, how great a velocity
must be impaired to a body in order
to carry II beyond the control OF the
earth's gravitation without any re-
gard TO the gravitation Of other
bodies in the universe). is, put from
mathematical symbols into words:
Velocity equals square root of the
product obtained by multiplying the
radius of the earth by 2g.
You will remember that e rep
resents the acceleration OF gravity
at the earth's surface, or about 1922
feet per second per second. Then
2g equals 644. the number by which
you must multiply the radius OF the
earth, which must also be express
ed in feet
Now. the radius Of the earth in
s.o4 4. oLo,,+ on Q0O mn oud this