The stages of a two-stage rocket have initial masses and respectively and carry a payload of mass . Both stages have equal structure factors and equal relative exhaust speed . The rocket mass, is fixed and .
According to multi-stage rocket’s flight equation (see “Viva Rocketry! Part 2“), the final speed of a two-stage rocket is
Let , it becomes
where . We will maximize with an appropriate choice of .
That is, given
where . Maximize with an appropriate value of .
The above optimization problem is solved using calculus (see “Viva Rocketry! Part 2“). However, there is an alternative that requires only high school mathematics with the help of a Computer Algebra System (CAS). This non-calculus approach places more emphasis on problem solving through mathematical thinking, as all symbolic calculations are carried out by the CAS (e.g., see Fig. 2). It also makes a range of interesting problems readily tackled with minimum mathematical prerequisites.
The fact that
is a monotonic increasing function
(1) can be written as
for gives if .
Hence, (1) is a quadratic equation. For it to have solution, its discriminant must be nonnegative, i.e.,
If , (3) is a quadratic equation.
Solving (3) yields two solutions
and, the solution to (2) is
We prove that (4) is true by showing (5) is false:
It can be written as
Since (see Exercise 1) and,
solve (7) for yields
It follows that for .
Consequently, is a negative quantity. i.e.,
which tells that (5) is false.
Hence, when , the global maximum is .
Solving for :
attains maximum at .
In fact, attains maxima at even when , as shown below:
Solving for , we have
Only is valid (see Exercise-2),
Solve quadratic equation for yields
The coefficient of in is , a negative quantity (see Exercise-3).
The implication is that is a negative quantity when .
Hence, (8) is a positive quantity, i.e.,
We therefore conclude
attains its maximum at .