Research on rocket flight performance has shown that typical single-stage rockets cannot serve as the carrier vehicle for launching satellite into orbit. Instead, multi-stage rockets are used in practice with two-stage rockets being the most common. The jettisoning of stages allows decreasing the mass of the remaining rocket in order for it to accelerate rapidly till reaching its desired velocity and height.
Optimizing flight performance is a non-trivial problem in the field of rocketry. This post examines a two-stage rocket flight performance through rigorous mathematical analysis. A Computer Algebra System (CAS) is employed to carry out the symbolic computations in the process. CAS has been proven to be an efficient tool in carry out laborious mathematical calculations for decades. This post reports on the process and the results of using Omega CAS explorer, a Maxima based CAS to solve this complex problem.
A two-stage rocket consists of a payload propelled by two stages of masses (first stage) and (second stage), both with structure factor . The exhaust speed of the first stage is , and of second stage . The initial total mass, is fixed. The ratio is small.
Based on Tsiolkovsky’s equation, we derived the multi-stage rocket flight equation . For a two-stage rocket, the final velocity can be calculated from the following:
Let , so that (1) becomes
We seek an appropriate value of that maximizes .
Consider as a function of , its derivative is computed (see Fig. 1)
We have .
That is, .
As shown in Fig. 2, can be expressed as
Notice that .
Solving for gives two solutions (see Fig. 3)
We rewrite the expression under the square root in and as a quadratic function of : and compute (see Fig. 4)
If , . It implies that is positive since . When , where is still positive since as a result of , the zero point of function is .
The expression under the square root is positive means both and are real-valued and (see Fig. 5), i.e., .
From (3) where , we deduce the following:
For all , if then
For all , if then
For all , if then
Moreover, from Fig. 6,
Since the expression in the numerator of , namely
It follows that
The implication is that has at least one zero point between and .
However, if both and , the two known zero points of are between and , by () and (), must be positive, which contradicts (4). Therefore, must have only one zero point between and .
We will proceed to show that the only zero lies between and is .
There are two cases to consider.
Case 1 () since and . But this contradicts (). Therefore, must not be positive.
Case 2 () The denominator of is negative since . However, , the terms not under the square root in the numerator of can be expressed as . This is a positive expression since implies that . Therefore, .
The fact that only lies between and , together with () and () proves that is where the global maximum of occurs.
can be simplified to a Taylor series expansion (see Fig. 7)
The result produced by CAS can be written as . However, it is incorrect as would suggest that is a negative quantity when is small.
To obtain a correct Taylor series expansion for , we rewrite as first where
Its first order Taylor series is then computed (see Fig. 8)
The first term of the result can be written as . Bring the value of into the result, we have:
To compute from (6) , we substitute for in (2) and compute its Taylor series expansion about (see Fig. 9 )
Writing its first term as and substituting the value yields:
It is positive when is small.
We have shown the time-saving, error-reduction advantages of using CAS to aid manipulation of complex mathematical expressions. On the other hand, we also caution that just as is the cases with any software system, CAS may contain software bugs that need to be detected and weeded out with a well- trained mathematical mind.
M. Xue, Viva Rockettry! Part 2 https://vroomlab.wordpress.com/2019/01/31/viva-rocketry-part-2
Omega: A Computer Algebra System Explorer http://www.omega-math.com