FIRST STEPS TOWARD SPACE - Smithsonian Institution Libraries

1
Some Jet Propulsion Formulas of Over Thirty Years Ago
In two articles published in L'Aerotecnica in 1933
and 1934, 1 some formulas were presented, together
with diagrams concerning the vertical motion of a
vehicle having constant acceleration and constant
exhaust velocity.
It is very interesting to note that the braking
effect produced by air on a rocket in vertical motion,
calculated by the formulas in the above 1934
articles, coincides perfectly with results presented
for the U.S. Navy Neptune (Viking) sounding
rocket in a 1949 English publication. 2
Vertical Motion by Constant Acceleration
The first formulas concern the vertical motion of
a space ship with a regulated jet for maintaining a
constant acceleration w; with a given constant exhaust
velocity of the jet v; with the earth considered
fully spherical with a radius r; and with no allowance
for air friction.
The variation of mass m in the time t is given by
the following formula:
, . ,, w 9.81i /
logm = logM0-—t /
r
|_arctan(t|/f)
° cv CV \ 2^7
1
+-„sin 2 arctan
where M0 is the initial value of mass and c —
2.30259 is the constant for conversion of a natural
logarithm into a common logarithm. The altitude x
at which the motor must be turned off for reaching
the altitude h is given by the relation
2 ivx = 19.62 r 2 (—
\x + r
-j-^—X
h + r /
(2)
and, in particular, the altitude x at which the motor
ALDO BARTOCCI, Italy
must be turned off to escape from terrestrial gravity
is given by the formula
2wx = 19.62 x + r
Applying the formula (3) to some numerical examples,
it appears that the initial mass necessary
for a given excursion diminishes with increasing
exhaust velocity of the jet, thus producing higher
acceleration values.
The air resistance in kilograms, which has not
been taken into consideration in formula (1), is
given by the formula:
R = F(V)'d-a*
1000 •*'
where F(V) is a function depending on the speed of
vehicle V; d is the density of the air; a is the diameter,
in meters, of the transverse section of the
vehicle; and i is a shape coefficient.
Assuming for the function F(V) the values in
function of speed adopted by ballistics, we obtain
for air resistance the curves shown in Figure 1
correlating time and acceleration.
Vertical Motion by Constant Efflux
The second group of formulas concern vertical
motion of an unmanned rocket with a constant
thrust (constant mass efflux and constant exhaust
velocity).
Not considering air friction and considering constant
acceleration of gravity during the operating
time of the motor, we have formulas in which
M0 is the initial mass of rocket, Mt is the final mass
at the end of the combustion, and n is the ratio be-
(3)
(4)