Fig. 1

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

where

or

(1) can be written as

where

,

.

Since means

.

That is

.

Solve

for gives if .

Hence, (1) is a quadratic equation. For it to have solution, its discriminant must be nonnegative, i.e.,

Consider

If , (3) is a quadratic equation.

Solving (3) yields two solutions

,

.

Since ,

(4) implies

and, the solution to (2) is

or

i.e.,

or

We prove that (4) is true by showing (5) is false:

Consider :

where

.

It can be written as

where

,

,

.

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 :

,

we have

.

Therefore,

attains maximum at .

In fact, attains maxima at even when , as shown below:

Solving for , we have

or .

Only is valid (see Exercise-2),

When ,

where

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 .

Fig. 2

Exercise-1 Prove:

Exercise-2 Prove:

Exercise-3 Prove: