Nomenclature

C _d

drag coefficient

C _dy

side drag coefficient

C _l

lift coefficient

D

drag force [N]

D _y

side drag force [N]

F

net force [N]

F _b

aerodynamics force [N]

F _g

gravitational force [N]

F _t

thrust force [N]

g

gravitational acceleration [m/s²]

g ₀

gravitational acceleration at sea level [m/s²]

H

angular momentum [kg-m²/s]

I _sp

specific impulse [s]

k

specific heat ratio

L

lift force [N]

m

flight vehicle mass [kg]

m ₀

rocket initial mass [kg]

m _p

rocket propellant mass [kg]

M

net torque [N-m]

MR

mass ratio

p

angular velocity for roll axis [rad/s]

P _c

chamber pressure [Pa]

P _e

ambient pressure [Pa]

q

angular velocity for pitch axis [rad/s]

r

angular velocity for yaw axis [rad/s]

R

gas constant [J/(kg-K)]

S

cross sectional area in axial direction [m²]

S _y

cross sectional area in transverse direction [m²]

T

magnitude of thrust [N]

T _c

chamber temperature [K]

v _e

effective exhaust velocity [m/s]

$\vec V$

velocity vector with respect to the inertial frame [m/s]

V _t

total velocity of the body fixed frame [m/s]

$\vec w$

relative angular velocity vector between the body fixed frame and the inertial frame [rad/s]

x _cp

static margin [m]

$\alpha $

angle of attack [degree]

$\beta $

angle of side slip [degree]

ζ

propellant fraction

$\varTheta$

Euler angle for pitch axis [rad]

$\rho $

air density [kg/m³]

$\varphi$

Euler angle for roll axis [rad]

$\varPsi$

Euler angle for yaw axis [rad]

2.0 Maximisation of a final altitude of an ascending rocket: the goddard problem

The problem of maximising a final altitude of a vertically launched rocket can be traced back to 1919 when Robert H. Goddard [Reference Goddard13] attempted to find the optimum thrust curve for a given amount of propellant. The eponymous Goddard problem was approached using different mathematical methods, including the calculus of variations at an earlier time and a few assumptions on the equations of motion, circumventing difficulties in solving the governing equation analytically, before more constraints were considered later, followed by numerical analysis and optimal control theory studies. The problem has been addressed extensively with reference to additional constraints such as constrained time of flight [Reference Tsiotras and Kelley14], a drag-law and its effects [Reference Tsiotras, Kelley and Kelley15], a dynamic pressure limitation [Reference Seywald and Cliff16], bounded thrust [Reference Munick19], and both dynamic pressure and thrust constraints [Reference Graichen and Petit17].

Goddard [Reference Goddard13] proposed the problem of maximising the altitude of a rocket in a vertical flight with a given amount of propellant, which was akin to minimising the required propellant mass to transfer a rocket to an assigned maximal height, finding an optimised thrust composed of a piecewise continuous thrust function of time. He identified the problem as an unsolved problem of the calculus of variations and suggested no further formulation or solution. A few years later, Hamel [Reference Hamel20] proved the existence of a solution by means of the calculations of variations, with an assumption of a linear rocket mass variation. More assumptions were considered in the following studies on the Goddard problem.

Malina et al. [Reference Malina and Smith21] considered a rocket launched vertically in a vacuum and in a resisting medium condition. In order to show maximum height as a function of three variables they mainly explored: effective exhaust velocity, mass ratio, and initial acceleration. They discovered that without consideration of an aerodynamics drag force, it is best to use the propellant in the shortest possible time reaching maximum velocity immediately, based on a numerically estimated reachable altitude that increased with increasing initial acceleration. However, with an aerodynamics drag, high initial velocity had a negative impact on peak altitude maximisation due to the high aerodynamics drag that tends to occur at a low altitude, where the air is relatively denser than that of a higher altitude. Thus, in such cases, a prolonged powered-flight was beneficial to get over the hump of the drag curve, allowing a rocket to reach a higher altitude. In other words, there was a certain initial acceleration that is optimal for a maximum altitude, which was later found again by Ivey et al. [Reference Ivey, Bowen and Oborny22], with a range of one to three times gravitational acceleration.

Tsien et al. [Reference Tsien23] provided an extensive numerical analysis of flight trajectories to show optimum thrust programming by considering a velocity dependent linear drag and a quadratic drag. With a varying drag model, it was found that it is more advantageous to have a lower velocity at low altitudes in order to increase the altitude at the end of the propellant burning cycle. For the optimum thrust programming, after the initial impulse, the amount of thrust should increase with altitude in general. In addition, as the linear drag does not increase with velocity as quickly as the quadratic drag, the linear drag favours a shorter burn time and high velocity with large initial acceleration to obtain a higher reachable altitude, while the drag from the quadratic model favours lower velocity and longer burn time.

Bryson Jr et al. [Reference Bryson and Ross24] devised a method to handle a non-increasing rocket mass constraint and some numerical calculations, resulting in optimum trajectories that took into account an aerodynamic drag. During its three flight stages, composed of an initial instantaneous boost, a sustain phase, and a coasting phase, the optimised initial flight path angle was 69 degrees ground-to-ground for the maximum range, the value of which would have been 45 degrees in the absence of drag. The study examined the difference between trajectories with an aerodynamic drag and without.

Garfinkel [Reference Garfinkel25] tried to broaden the number of soluble cases rather than focusing on a select few specific ones allowing previous solutions to meet the monotone rocket mass requirement, which had caused the problem at that time. The study solved the thrust optimising problem of a vertically ascending rocket, assuming an isothermal atmosphere and admissibility of an infinite thruster, in two specific cases: 1) Mach number and fuel supply are sufficiently large, and 2) the drag is a convex function of velocity. The first scenario embraced very general physical drag functions and was valid for Earth, while the second was extended to all atmospheric conditions including extra-terrestrial with limited applicability to drag functions, which was reasonably common at that time, in order to make the Goddard problem solvable.

Munick [Reference Munick19] considered a bounded thrust, identifying an optimised thrust programming that consisted of a piecewise continuous thrust as a function of time to maximise its peak altitude. It was concluded that when using the thrust sufficiently high in a finite level, the optimal control contains three subarc functions at most. The results of Munick’s research extended the findings of previous studies to a larger range of drag functions, including a drag range near the speed of sound.

Seywald’s study [Reference Seywald and Cliff16,Reference Seywald and Cliff26] was concerned with the effect of two additional conditions as constraints: a dynamics pressure limit and a specified final time. The effect of those restrictions on an optimal thrust switching structure was investigated and a thrust structure was identified for a maximum attainable altitude using nine different types of thrust arcs. The constrained time of flight was also one of the conditions considered for the Goddard problem and the maximisation of altitude and for a given flight time in Panagiotis’s study [Reference Tsiotras and Kelley14] where maximising the final altitude of a vertically launched rockets has been analysed in regard to quadratic draw law and bounded thrust. Seywald’s work was then further explored by Graichen and Petit [Reference Graichen and Petit17] concerning dynamics pressure and thrust constraints using a different methodology. By using a saturation function, it systematically incorporated the constraints into the equation and resulted in an unconstrained optimal control problem that required no knowledge of the thrust switching structure, implying its broader applicability to various constraints.

All of these efforts, which have continued until the last decade, have largely focused on the shape of the aerodynamic drag function, considering it one of the major concerns in terms of making the equation of motion more accommodating when approaching the Goddard problem. There has also been an interest in additional constraints for optimisation, including dynamic pressure limitation, atmospheric density, a limited time of flight, thrust constraints, and optimum thrust programming, implying several assumptions were beneficial in regard to the mathematical models and methods adopted in the previous theoretical studies. The results from the theoretical work on optimum conditions represent a useful guide for rocket operational conditions in that they detail the influence of several design and operational parameters on rocket flight performance. However, although the preliminary analysis is informative, to some extent, in designing a rocket, it does not necessarily guarantee that the operational conditions suggested are physically feasible, as the analysis was approached mathematically with less emphasis on the practicability of the system configuration from a rocket design point of view. Therefore, in relation to optimal flight, further research on design parameters is required with a greater emphasis on their practical properties which are conducive to peak altitude maximisation. This is the main concern and objective of this work, which will propose a new design approach along such lines, detailed in the following sections.

3.0 New design approach to altitude maximisation

With regards to an operational design conducive to altitude maximisation, numerous analytical estimation studies and simulation-based approaches have considered the governing equation of motion, producing a range of results. There were several assumptions as to how the analytic equations could be solved, but with limited practicality in real rocket design applications. Simulation-based studies also suggested certain optimisation conditions, but these are arguably too specific to be applied to other rocket designs, and even if a broader range of design variables were considered, they could not be applied pragmatically to an effective rocket design process. Thus, whether the selected design parameters are appropriate determines whether or not the optimisation results are practically applicable to a sounding rocket design process. Despite the numerous variables previously reported, there is also room for more intuitive data on design inputs and their effect on optimised flight performance. More studies are required on this perspective to obtain clearer outputs versus rocket system design inputs (i.e. reachable altitudes optimised with respect to the main rocket design variables).

The principal design parameters of a sounding rocket should be properly determined and considered in optimisation work, not only to provide intuitive and broad rocket design domains which optimise peak altitude, but also to make the optimisation data meaningful in terms of practicability. Thus, considerable effort was made in this work to find both useful, effective and viable design parameters which provided an optimised design domain reference in the design of a sounding rocket, particularly a single staged type.

In reality, as most engines are developed with a certain amount of targeted thrust, while some are available off-the-shelf with certain performance specifications, the magnitude of thrust is one of the most practical design parameters, and it must be considered in the sounding rocket design process. Engine selection or development is generally followed by a system design process that includes a system configuration in relation to pipelines, tanks, valves, casing, payload and electrical systems for avionics, data collection and communication. In terms of practicality, therefore, a burn time is determined by choosing a type of tank and volume which depends on total propellant mass and the maximum combustion duration. Then the mass ratio of the propellant and the rest of the rocket system (dry mass) can be estimated. Thus, primary major design parameters for rocket systems should include a magnitude of thrust, propellant type, burn time and propellant fraction (i.e. mass ratio).

Some might argue that more concern should be accorded to chamber pressure, specific impulse, and propellant flow rate, the importance of which might appear to have been neglected in this study. However, these variables are related to each other and they were considered as dependent variables in this work. For example, following the thrust equation as shown in Equation (1), once the amount of thrust has been determined, the required propellant consumption rate can be calculated considering specific impulse performance.

(1) \begin{align}T{\rm{\;}} = {\rm{\;}}{\dot m_p}{I_{sp}}{g_0}\end{align}

T is the magnitude of thrust, ${\dot m_p}$ propellant mass flow rate, I _sp specific impulse and g ₀ gravitational acceleration at sea level. Specific impulse (I _sp ) performance is, as shown in Equation (2), a function of propellant type (specific heat, gas constant and combustion temperature) and working pressure ratio.

(2) \begin{align}{I_{sp}} = \frac{{{v_e}}}{{{g_0}}}\;\;\sim \;\frac{1}{{{g_0}}}\sqrt {\frac{{2k}}{{k - 1}}R{T_c}\left[ {1 - {{\left( {\frac{{{P_e}}}{{{P_c}}}} \right)}^{\left( {k - 1} \right)/k}}} \right]} \end{align}

v _e is the effective exhaust velocity, g ₀ gravitational acceleration at sea level, k specific heat ratio, R gas constant, T _c chamber temperature, P _c chamber pressure and P _e ambient pressure. As there is a common range of design values for chamber pressure (P _c ) to be achieved in a rocket chamber, once propellant type (and thus its specific heat ratio, gas constant, and theoretical combustion temperature, from a chemical equilibrium analysis) is decided, accordingly specific impulse is expected to be in a certain range, and it can be assumed to be determined by propellant type. This is also a reasonable assumption because the chamber pressure effect on specific impulse is not linear. Once the operation chamber pressure reaches a certain amount, its further effect on specific impulse becomes marginal, as seen in Equation (2) for I _sp with respect to the P _c . Typically, chamber pressure of 20 – 30 bar or more [Reference Chern, Wu, Chen and Wu5,Reference Huh, Ahn, Kim, Song, Yoon and Kwon27–Reference Brooks, Pitot de la Beaujardiere, Chowdhury, Genevieve and Roberts30] is considered depending on the mission, but it is typically in the range of several tens of bars for the general chamber design pressure of sounding rockets. Therefore, it is arguably reasonable to assume that specific impulse performance is determined by propellant type. Also, given the potentially limited range of those parameters, they were not the major independent variables considered in this study.

Another aspect making this effective design variables selection more meaningful is that there is a trend in engineering toward sounding rocket sizing, depending on the amount of thrust. For example, as can be seen in the specifications of sounding rockets from previous studies in Table 1, and thrusts and body diameters from single-stage sounding rockets in Fig. 1, despite the differences between the propellant types used, there is a link between sounding rocket body diameter size and the amount of thrust targeted. With regards to this physical aspect of engineering, a flight simulation can be simplified, at least with respect to a drag force, which is proportional to the cross section area of a rocket body. On the basis of this empirical data, the drag force was determined, according to the magnitude of thrust produced by a sounding rocket.

Table 1.

Sounding rocket specifications from previous studies

Figure 1.

Thrusts and body diameters mainly of single-stage sounding rockets reported in previous studies (from Table 1).

As a result, among the numerous design variables that can be examined as input variables, this study identified three main design parameters which can be applied to a sounding rocket design process with a focus on reachable altitude maximisation: Magnitude of thrust, burn time, and propellant fraction (i.e. mass ratio). The usefulness and/or effectiveness of the selected input variables were expected to be verified in this study, resulting in a new type of design domain with broad and intuitive flight performance characteristics which would represent a design reference for a sounding rocket to reach a target altitude.

4.0 Flight trajectory simulation

In order to investigate the influence of the selected design variables on flight performance and suggest appropriate design domains, a flight trajectory was simulated. A six-degree-of-freedom flight trajectory in-house code was developed and a shooting method was employed. The shooting method is a method used in numerical analysis to find a solution trying different initial conditions until a satisfactory result is obtained. The method was evaluated as suitable for this simulation, as during the solving process it would plot a simulation result for each initial condition, providing a design domain with an optimised design reference.

For the trajectory calculation, the equation of rotational motion, kinematic relation and Newton’s second law of motion were considered as the main governing equations. Additionally, the quaternion equation was utilised to circumvent a singular point while solving Euler angles. As Newton’s equation of motion applies to an inertial frame, two frames were used; body fixed frame and Earth fixed frame. The body fixed frame was a frame fixed to a rocket, more specifically on the point of the body where the centre of gravity was located, which moves and rotates together with the rocket body during a translational and rotational motion, and thus it was not an inertial frame. For the Earth fixed frame, North-East-Down (NED) frame was used, which was located at the surface of the Earth and assumed as an inertial frame without consideration of revolution and rotation of the Earth.

For a translational motion, as shown in Equation (3), Newton’s second law of motion was applied to the inertial frame and then expressed with a velocity of the body fixed frame, ${\left({\frac{{d\overrightarrow{\textbf{V}}}}{{dt}}}\right)_r}$ , and relative angular velocity between the body fixed and the Earth fixed frame, $\vec w$ , where m is a rocket mass and t is time.

(3) \begin{align}\sum \overrightarrow{\textbf{F}} = m\frac{d}{{dt}}\left( {\overrightarrow{\textbf{V}}} \right) = m\left[ {{{\left( {\frac{{d\vec V}}{{dt}}} \right)}_{\!\!r}} + \overrightarrow{\textbf{w}} \times \overrightarrow{\textbf{V}}} \right]\end{align}

Thrust, aerodynamic force and gravitational force were taken into consideration for the net force, $\Sigma \overrightarrow{\textbf{F}}$ . Aerodynamic force at the body fixed frame for each axis was, as shown in Equation (4), a function of angle of attack α, side slip $\rho$ , air density $\rho$ , total velocity V _t , drag coefficient C _d and C _dy , lift coefficient C _l , and cross section area S and S _y .

(4) \begin{align}{\overrightarrow{\textbf{F}}_{\textbf{b}}} = \left[ {\begin{array}{*{20}{c}}{{F_{bx}}}\\{{F_{by}}}\\{{F_{bz}}}\end{array}} \right]{\rm{\;}} = {\rm{\;}}\left[ {\begin{array}{*{20}{c}}{ - D\cos \alpha + L\sin \alpha }\\{{D_y}}\\{ - D\sin \alpha - L\cos \alpha }\end{array}} \right]\end{align}

where

(5) \begin{align}D = \frac{1}{2}\rho {\left( {V_t\cos \beta } \right)^2}S{C_d}\end{align}

(6) \begin{align}L = \frac{1}{2}\rho {\left( {{V_t}\cos \beta } \right)^2}S{C_l}\end{align}

(7) \begin{align}{D_y} = \frac{1}{2}\rho {\left( {{V_t}\sin \beta } \right)^2}{S_y}{C_{dy}}\end{align}

For gravitational force, Euler angles, $\varPsi$ , $\theta$ , and $\varphi$ , were required to identify the rocket attitude and determine the gravitational force distribution to each axis, mg _x , mg _y , and mg _z .

(8) \begin{align}{\overrightarrow{\textbf{F}}_{\textbf{g}}} = m\overrightarrow{\textbf{g}} = m\left[ {\begin{array}{*{20}{c}}{{g_x}}\\{{g_y}}\\{{g_z}}\end{array}} \right]{\rm{\;}} = {\rm{\;}}{\varphi ^{ - 1}}{\theta ^{ - 1}}\left[ {\begin{array}{*{20}{c}}0\\0\\{mg}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{ - mg\sin \theta }\\{mg\cos \theta \sin \varphi }\\{mg\cos \theta \cos \varphi }\end{array}} \right]\end{align}

Thrust was assumed to be produced in the axial direction only without consideration of thrust vector control as described in Equation (9).

(9) \begin{align}{\overrightarrow{\textbf{F}}_{\textbf{t}}} = \left[ {\begin{array}{*{20}{c}}{{F_{tx}}}\\{{F_{ty}}}\\{{F_{tz}}}\end{array}} \right]{\rm{\;}} = {\rm{\;}}\left[ {\begin{array}{*{20}{c}}{thrust}\\0\\0\end{array}} \right]\end{align}

Then Equation (3) for Newton’s second law gave information on the body fixed accelerations ( $\dot u$ , $\dot v$ , $\dot w$ ), with the input of aerodynamic force, thrust, gravitational force, mass, velocities (u, v, w) and angular velocities for each axis (p, q, r). Likewise, the rotation motion was calculated by solving the equation of Newton’s second law of rotation, as shown in Equation (10) with inputs of rocket moment of inertias, torques, and angular velocities, for outputs of angular accelerations.

(10) \begin{align}\sum \overrightarrow{\textbf{M}} = \frac{{d\overrightarrow{\textbf{H}}}}{{dt}} = {\left( {\frac{{d\overrightarrow{\textbf{H}}}}{{dt}}} \right)_r} + \overrightarrow{\textbf{w}} \times \overrightarrow{\textbf{H}}\end{align}

Torque on each axis ( $\bar L$ , $M$ , $N$ ) was assumed to be applied on a rocket, the cause of which was exclusively the aerodynamic forces acting on the centre of pressure, and they were functions of the static margin x _cp (a distance between the centre of pressure and the centre of gravity), lift L, drag D and D _y , and angle of attack $\;\alpha $ .

(11) \begin{align}\overrightarrow{\textbf{M}} = \left[\!\!{\begin{array}{*{20}{c}}{\bar L}\\M\\N\end{array}} \right]{\rm{\;}} = {\rm{\;}}\left[ {\begin{array}{*{20}{c}}0\\{ - {x_{cp}}\left( {L\cos \alpha + D\sin \alpha } \right)}\\{ - {x_{cp}}{D_y}}\end{array}} \!\!\right]\end{align}

The attitude of a rocket can be determined by using the Euler angles, solving the kinematic relation below.

(12) \begin{align}\left[ {\begin{array}{*{20}{c}}{\dot \varphi }\\[3pt] {\dot \theta }\\[3pt] {\dot \Psi }\end{array}} \right] = \left[ {\begin{array}{*{20}{c@{\quad}c@{\quad}c}}1 & {\sin \varphi \tan \theta } & {\cos \varphi \tan \theta }\\[3pt] 0 & {\cos \varphi } & { -\sin \varphi }\\[3pt] 0 & {\sin \varphi \sec \theta } & {\cos \varphi \sec \theta }\end{array}} \right]\left[ {\begin{array}{*{20}{c}}p\\[3pt] q\\[3pt] r\end{array}} \right]\end{align}

There is, however, a singular point for tan $\;\theta$ when $\theta$ is 90^°, which is a common occurrence in a rocket flight simulation. Thus, a quaternion equation was used instead to circumvent the singularity. The four quaternion variables were acquired by solving additional quaternion equations, detailed equations of which can be found in [Reference Yang50], and they were converted to Euler angles, $\varPsi$ , $\theta$ , and $\varphi$ , using the Euler angle-quaternion conversion equation.

These three main equations: translational motion, rotational motion, and quaternion equations, were numerically solved via a specific process. The rotational motion was first calculated to determine the angular acceleration of a rocket at the next time step after an initial condition, and then the quaternion equation was solved resulting in the Euler angles and the rocket attitude, with which the gravitational force distribution to each axis of the body fixed frame was identified. After determining the net force acting on the rocket, a translational motion was calculated, acquiring accelerations of the rocket on each axis. The three ordinary differential equations were numerically solved using the Runge Kutta 4^th order method at each time step for the entire flight time. The acquired variables at the body fixed frame were then converted to variables at the Earth fixed frame using the quaternion variables. Finally, a rocket flight trajectory was drawn by integrating accelerations and then velocities for each axis of the Earth fixed frame.

The flight simulation was conducted on the basis of certain conditions and assumptions. As the NED coordinate was used as an inertial frame, it meant it was relevant to a flight problem on flat Earth (neglecting the rotation of Earth), an appropriate assumption for a sounding rocket which has a shorter flight range than that of a launch vehicle. The variation of gravitational acceleration with altitude was also neglected, which has less than a 3% error at a 100 km altitude and less than 6% at 200 km. The US standard atmospheric data was employed for air density at each altitude. The sounding rocket was assumed to have a rigid body with a constant rate of propellant consumption and to be devoid of thruster vector controlling or throttling. Although the varying drag coefficient can be numerically estimated and compared with other sources, such as OpenRocket [Reference Niskanen51] or RasAero [Reference Rogers and Cooper52]; semi-empirical data such as Missile Datcom [Reference Burns, Deters, Stoy, Vukelich and Blake53]; or via a wind tunnel test, a constant value was used in this simulation, not only to achieve simplicity, but also to establish a common reference and make it easier to draw comparisons with other various rocket designs of different aerodynamic shapes. The constant drag coefficient for the simulation was 0.35 and the lift coefficient was zero.

As described earlier, this work focused on three main design parameters for a trajectory simulation. Propellant mass fraction, ζ, was one of the three parameters, which is defined as shown in Equation (13), where m _p is propellant mass and m ₀ is the total mass of the vehicle.

(13) \begin{align}\zeta = \frac{{{m_p}}}{{{m_0}}}\end{align}

In this study, the term mass ratio (MR) was used to refer to the propellant fraction, the ratio of propellant to the total mass of the sounding rocket. For example, 0.7 of MR in this study meant propellant accounted for 70% of the total rocket mass, with 30% (dry mass) composed of the rocket systems, such as structures, payloads and any subsystem except the propellant.

For the flight simulation, the mass ratio (propellant fraction) in the range of 0.1 – 0.7 was covered. In view of the increasing use of hybrid rockets in sounding rocket flight testing, all simulated rockets had specific impulse (Isp) of 230 s, which is the typical Isp performance of a hybrid rocket. The amount of thrust simulated was in the range of 5,000 N to 35,000 N and it was a constant thrust profile for each simulation case during the burn time considered. As the diameter of a rocket’s body generally increases with thrust, as shown in Fig. 1, the different thrust performance inputs in each simulation was taken into account with different rocket diameters. It was 25 cm for 5,000 N, 30 cm for 10,000 – 15,000 N, 35 cm for 20,000 – 25,000 N, and 40 cm for 30,000 N.

Burn time covered in the simulation was 50 s. The launch angle was 85 degrees and the static margin, a distance between the centre of gravity and the centre of pressure, was determined as 1.5, which is known as an appropriate value for rocket stability based on previous studies [Reference Huh, Ahn, Kim, Song, Yoon and Kwon27,Reference Barrowman54]. The flight simulation conditions are listed in Table 2. With these input variables and ranges, a flight trajectory was simulated for each different combination of these design parameters, and the peak altitudes were ascertained. For example, Fig. 2 shows the peak altitudes collected with respect to burn time and propellant fraction, MR, for a 15,000 N class single-stage sounding rocket.

Table 2.

Flight simulation conditions for a single-stage rocket

Figure 2.

Reachable altitudes of a 15,000 N class sounding rocket with respect to burn time and mass ratio (MR, i.e. propellant fraction).