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 speeds. If the rocket mass, , is fixed, show that the condition for maximal final speed is
Find the optimal ratio when .
According to multi-stage rocket’s flight equation (see “Viva Rocketry! Part 2“), the final speed of a two stage rocket is
Let , we have
Differentiate with respect to gives
It follows that implies
That is, . i.e.,
It is the condition for an extreme value of . Specifically, the condition to attain a maximum (see Exercise-2)
When , solving
yields two pairs:
Only (3) is valid (see Exercise-1)
The entire process is captured in Fig. 2.
Exercise-1 Given , prove:
Exercise-2 From (1), prove the extreme value attained under (2) is a maximum.
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 .
A rocket with stages is a composition of single stage rocket (see Fig. 1) Each stage has its own casing, instruments and fuel. The th stage houses the payload.
The model is illustrated in Fig. 2, the stage having initial total mass and containing fuel . The exhaust speed of the stage is .
The flight of multi-stage rocket starts with the stage fires its engine and the rocket is lifted. When all the fuel in the stage has been burnt, the stage’s casing and instruments are detached. The remaining stages of the rocket continue the flight with stage’s engine ignited.
Generally, the rocket starts its stage of flight with final velocity achieved at the end of previous stage of flight. The entire rocket is propelled by the fuel in the casing of the rocket. When all the fuel for this stage has been burnt, the casing is separated from the rest of the stages. The flight of the rocket is completed if . Otherwise, it enters the next stage of flight.
A rocket is programmed to burn and ejects its propellant at the variable rate , where and are positive constants. The rocket is launched vertically from rest. Neglecting all external forces except gravity, show that the final speed given to the payload, of mass , when all the fuel has been burnt is
Here is the speed of the propellant relative to the rocket, the initial rocket mass, excluding the payload. The initial fuel mass is .
Exercise 2. Before firing, a single stage rocket has total mass , which comprises the casing, instruments etc, with mass , and the fuel. The fuel is programmed to burn and to be ejected at a variable rate such that the total mass of the rocket at any time , during which the fuel is being burnt, is given by
where is a constant.
The rocket is launched vertically from rest. Neglect all external forces except gravity, show that the height attained at the instant the fuel is fully consumed is
It follows that the percentage reduction in the predicted final speed due to the inclusion of gravity is
Using the given values which are typical, the estimated value of (3) (see Fig. 2) is
This shows the results obtained without taking gravity into consideration can be regarded as a reasonable approximation and the characteristics of rocket flight indicated in “Viva Rocketry! Part 1” are valid.
Since , (1) can be written as
To find the distance travelled while the fuel is burnt, we solve yet another initial-value problem:
The solution (see Fig. 3) is
Hence, the height reached at the burnt out time is
Using the given values, we estimate that (see Fig. 4)
Exercise 1: Find the distance the rocket travelled while the fuel is burnt by solving the following initial-value problem:
Shown in Fig. 1 is an experimental car propelled by a rocket motor. The drag force (air resistance) is given by . The initial mass of the car, which includes fuel of mass , is . The rocket motor is burning fuel at the rate of with an exhaust velocity of relative to the car. The car is at rest at . Show that the velocity of the car is given by, for ,
where , and is the time when the fuel is burnt out.