Assigment 1: Rocket Science

While high budget endeavours such as manned or interplanetary space flight receive media headlines, much space research is actually done in more humble settings. Medium size rockets are used for low and medium altitude experiments. Centres of competence in Canada are at York and Calgary universities.

Likewise, manned interplanetary missions are still elusive. Instead research is done in so called "analogue environments", a fancy term for places which in suitable ways simulate conditions in space. The Mars analogue research station is set up in northern Canada on Devon island. There a model space station has been setup, a remote operated green house to feed the earth-bound "astronauts", and radio signals are used both for local communication and to call home (to CSA in Montreal) where a delay is added to the audio link just like if they had been on Mars.

In this assignment we will study a rocket model suitable for short and medium range flight on earth. Thus we will use a simple assumption of turbulent drag represented as a quadratic term. (A detailed understanding of Navier Stokes equations under turbulent flow is still elusive, and it was posed in year 2000 as one of seven Millenium Problems) You will learn how to rework the mathematical model into one suitable to supply to a numerical routine, and experiment with numerically computed trajectories.

For the assignment everyone is welcome to try all exercises and questions, but in particular:
4xx does Q1-4
6xx does Q1-8
All consider additional four bullet questions at end.

Rocket Model

A rocket at practical velocities can be modeled by Newton's laws. In particular, its motion is governed by the 2nd law:

\begin{displaymath}{\mathbf f}= {d \over dt}(m{\mathbf v}) \end{displaymath}

where
$m=m(t)$ is the mass. It is varying with time as fuel is burned.
${\mathbf v}$ is the velocity vector.
${\mathbf f}$ is the total force on the rocket. It consists of three parts:
  1. Engine thrust, assumed constant $T$ in the direction of motion as long as there is fuel, then 0.
  2. Gravitational force $gm = 9.8*m$ in the negative $y-direction$.
  3. Turbulent air drag $f_d = -{1\over 2}c\rho sv^2$, in the direction of motion, where $c$ is a shape dependent drag parameter (similar to $C_v$ for cars), $\rho$ is the air density, and $v$ is the scalar velocity magnitude ( $v = \vert\vert{\mathbf v}\vert\vert$).
Ignoring disturbances due to e.g. side winds, the above can be simplified into a system of 2 ODE in the horizontal and vertical x-y plane. Here \Theta is the angle of inclination w.r.t. the ground plane. For compactness we adopt the conventional notation for time derivatives ${dx \over dt} = \dot{x}$


$\displaystyle {\rm Horizontal \ force \ balance,}\ x:$ $\textstyle \cos \Theta (T-kv^2) = m \ddot{x} + \dot{m}\dot{x}$   (1)
$\displaystyle {\rm Vertical \ force \ balance,}\ y:$ $\textstyle \sin \Theta (T-kv^2) -mg = m \ddot{y} + \dot{m}\dot{y}$   (2)

where we have lumped in $k=-{1\over 2}c\rho s$.

Q1
To work with this using conventional ODE solution methods you need to re-express the system of 2nd order ODE as a system of 4 1st order ODE. Show your derivations.
Hint: notice $\dot{x} = v \cos \Theta$ and $\dot{y} = v \sin \Theta$, take derivatives once more. Substitute in above two ODE and rework/simplify into a system of ODE for $[\dot{x}, \dot{y},\dot{\Theta},\dot{v}]^T$

Numerical solution

The government of Alberta decides to fund the following medium size rocket:
$\displaystyle {\rm empty \ weight}$ $\textstyle m= 600kg$   (3)
$\displaystyle {\rm fuel \ capacity}$ $\textstyle m_f= 240kg$   (4)
$\displaystyle {\rm fuel \ burn \ rate}$ $\textstyle m_{prime}= -4{kg \over s}$   (5)
$\displaystyle {\rm Thrust}$ $\textstyle T = 68000 {kg \ m \over s^2}$   (6)
$\displaystyle {\rm Cross \ section}$ $\textstyle s= 1\rm m^2$   (7)

The University of Alberta administration is consulted and sets up a campus lab to measure:
$\displaystyle {\rm Air \ density }$ $\textstyle \rho=1.29 kg/m^3$   (8)
$\displaystyle {\rm Air \ resistance \ coe}$ $\textstyle c=0.2$   (9)

Furthermore a local contractor is employed to build a launch ramp, which launches the rocket at a 45 degree angle. (Why do you think they choose this?) Due to a confusion in geographical knowledge the initial experimental goal is to reach Devon, AB, 20km away to the south west. On the first launch, with the rocket fully tanked and the president of UA, provincial premier and other high brass present, not all goes as planned.
Q2
Show the trajectory and explain what went wrong
(An Internet search will show that many real world early rockets also failed, until scientists learned to properly understand and mathematically model the systems.)
Use the built in ode45 routine and one routine of your own design for the numerical integration. In the below a function rprime is defined as follows: 
  % rprime compute derivative of x,y,phi,v for rocket ODE at time t
  function yprime = rprime(t,y)

    yprime = zeros(4,1);
    <your computations>
    yprime(1) = 
    yprime(2) =
      :
To avoid numerical difficulties at the start an initial velocity of 1m/s is used (to think about: why).The coordinate system is in the origin. The initial conditions are then $y_0 = [0,0,pi/4,1]^T$. $t_0 = 0$ and a suitable duration has to be guessed, here 40s. ode45 is then called as follows: 
  [t,Y]=ode45('rprime',[0,40],[0,0,pi/4,1]');
  plot(Y(:,1),Y(:,2))
A fuel supply of 80kg remains. The rocket is relaunched with this, and results are slightly more encouraging.
Q3
Show also this trajectory and explain.

Study modifications

After public trust in the government and UofA administration has been exhausted, it is decided to give the CDE class a chance to correct the problems. A new amount of fuel is purchased and made available to the class.
Q4
Modify the fuel amount. What amount is needed to reach Devon?

Next however, local newspapers expose the mistake that the MARS analogue environment is not in Devon AB, but instead in Devon Nunavut.

Q5
What further modifications are needed to reach Devon, Nunavut?
Hint: Are starting angle and air density assumptions reasonable?

Further questions to research

What are some other issues in rocket modeling?
Q6
Is the assumption of constant thrust reasonable? When? Why?
Q7
Is the assumption of quadratic drag reasonable? When? Why?
Q8
What is the cruising altitude of a commercial jet airline, and why?
Numerical issues to think about:
Martin Jagersand, Computational Differential Equations course