Ideal rocket equation
I. Physical background
Assume a rocket of initial mass m0 and final mass
, with
being the total mass of fuel
carried by the rocket. The mass of the rocket changes according to the mass flow rate
. In other
words
,
. The mass is expelled with velocity
in a direction opposite to
the rockets flight path. Assume now that the rocket is moving at velocity v(t) and subject to some external
forces F (e.g. gravity, air resistance). Then, by conservation of momentum, we obtain:
Part 1:
If we assume that there are no external forces, then we obtain:
Solving this analytically, we arrive at the ideal rocket equation
with
( = initial velocity, = final velocity after all fuel has been expelled) [1]. If
then
is the final velocity of the rocket. Based on the desired value
and , the required amount of fuel in
terms of percentage of initial mass can then be calculated as
Part 2:
If we consider the rocket to be subjected to earth's gravity, i.e.
formula:
, then we obtain the following
II. Questions
Part 1
:
a) Start with
and arbitrary values for
,
and R. Calculate the change of the rocket's
velocity numerically for different values of
and compare the results with the ideal rocket
equation.
b) In order to reach a low earth orbit (LEO), we require a final velocity of about 9700m/s and using a
value of
and the analytical formula given above, this means that about 88.4% of
the rocket's initial mass have to be fuel (compare [1]). Using the same values for
and , use
numerical estimation of the rockets change in velocity in order to calculate the minimum required
fuel for several initial values of
and compare the ratios
to the value given above.
Part 2
:
Start with
and arbitrary values for
,
and R. Considering that the rocket is subjected to
gravity during the whole period of the flight, use numerical estimation of the rockets velocity in order
to calculate the maximal possible distance we can travel given a certain initial mass
.
III. Numerical calculation
Numerical simulation was done by the direct approach and in one dimension (z) only. Calculation were done
starting with full fuel and
, increasing in time steps of
at each iteration and continuing until the
fuel has been completely expelled. At each iteration, we start by updating the rocket's acceleration according
to
followed by an update of the velocity
, an update of the z-position
for the specific question) and finally an update of the rocket's mass
.
(if relevant
IV. Results and discussion
I started by numerically simulating the rocket's change of velocity over time either by ignoring gravity (figure
1A) or by incorporating it into the model (figure 1B).
Figure 1: Velocity of the rocket with respect to time. Numerical calculation starts at velocity
and full mass and
is iterated at time steps
until all fuel has been expelled.
;
;
. (A) The effect of gravity has been ignored; (B) Gravity has been added as
.
Unsurprisingly, the rocket that has not been subjected to gravity achieves a greater maximum velocity than
the rocket that has been slowed down due to gravity.
Part 1a:
In order to answer question 1a, I started with fixed values
;
;
,
then iterated through some values
and numerically calculated the maximum velocity (delta-v) achieved by
the rocket for each of these cases. The results are shown as the ratio
against delta-v (figure 2, compare
[1]) and as the numerical value of delta-v against the analytically calculated value for delta-v on the same
values
and
(figure 3).
Figure 2: Maximum velocity, delta-v, achieved by the rocket depending on the ratio
and ignoring gravity.
Figure 3: Scatter plot displaying the numerically estimated values of delta-v against the analytically calculated
values of delta-v for each set of values
and
. The red dotted line indicates the trend line fitted to the data and
the R value represents the Pearson correlation coefficient between the two datasets.
The trend of the graph shown in figure 2 is consistent with what has been shown analytically (compare [1])
and I also observed a perfect correlation (Pearson correlation coefficient of 1) between the numerically
estimated and analytically calculated values of delta-v on a number of values
and
. Thus, the
numerical simulation fits well with the analytical model.
Part 1b:
In order to solve question 1b, I started with fixed values
and
and then
iterated through some initial masses
. For each of these initial masses we then
estimate the minimum amount of fuel needed in order to achieve a final velocity delta-v of 9700m/s.
Specifically, for each mass
we start with an value
and then iterate. At each iteration we
calculate delta-v numerically. If delta-v is greater than 9700m/s, we exchange 100kg of fuel into 100kg of
payload. If delta-v is equal or smaller then 9700m/s, we stop and the last value of
is the minimum amount
of fuel required. The obtained ratios
are shown in figure 4.
Figure 4: The percentage of fuel (
) required in order to achieve a delta-v of 9700m/s with an initial
mass of
. The red dashed line indicates a value of 0.884 on the y-axis (
.
As can be seen from figure 4, the ratios
, which were estimated numerically, are highly consistent with
the analytically calculated ratio of 0.884, thus validating the numerical approach. In the lower range of
we
can however observe small deviations from the expected ratio of 0.884. This deviation is probably due to the
chosen values of R and the exact mass of fuel being exchanged at each iteration and could be adjusted for by
selecting adequate values for these parameters in lower ranges of .
Part 2:
In order to address question 2, I decided to numerically estimate the maximum distance that can be traveled
by a rocket depending on the initial mass
and the initial amount of fuel
and plot the distance against
the ratio
(figure 5). Simulations were done by starting with fixed values
;
;
and then iteration trough some values
and for each value
a calculation of
the maximal traveled distance as follows: For each
acceleration, velocity, z-position are updated with time
steps
if there is still fuel available or the velocity is still positive, i.e. the rocket is still traveling
upwards. If there is fuel available, the acceleration is calculated as
and the mass is updated at each
iteration as well. If there is no fuel available, the acceleration is just
and the mass is not updated any
longer. Note: in the start we have
, which means that the rocket should just remain at z=0 until we
have burned enough mass to obtain positive acceleration.
V. Implementation
Algorithms for numerical simulation and for plotting the figures have been implemented in MATLAB.
VI. References
[1]
Tsiolkovsky rocket equation. (2012, September 7). In Wikipedia, The Free Encyclopedia. Retrieved
16:44, September 17, 2012,
from http://en.wikipedia.org/w/index.php?title=Tsiolkovsky_rocket_equation&oldid=511262091