1.
A Model Rocket Flight Revisited
Anthony Gnas
Karly Kasten
Brandon VanderHeyden
Arianna Ziemer
Week #8­9
ME300 Fall 2015
Section 011
Professor V. Prantil
11/04/15

2. Problem Statement:
Three cases regarding a model rocket's flight were compared:
CASE I ­ CONSTANT THRUST, CONSTANT MASS
CASE II ­ VARIABLE THRUST, CONSTANT MASS
CASE III ­ VARIABLE THRUST, VARIABLE MASS
All three cases consist of two segments, first, the rocket was propelled upward by
thrust, opposing gravity and aerodynamic drag forces.Secondly, the rocket fuel was
exhausted and there was no longer thrust.The rocket continued to ascend, while
slowing down due to gravity and aerodynamic drag until it reached a peak altitude. In
Case I, which was investigated in Lab #1, thrust was given as a constant 3.78 Newton's
lasting 0.55 seconds, and the mass was given as a constant 71.1g. In Case II, thrust
was varied, given by experimental data. Mass was given as the same constant 71.1g.
Finally, in Case III, thrust was varied, again given by the experimental data, and this
time mass was also varied, using the fact that the mass burn­off was proportional to the
net impulse delivered by the fuel at the current point. These three cases were compared
at time steps of 0.05 seconds and 0.01 seconds. Additionally, the following values were
given:
ass of rocket with fuel 1.1gm = m = 7 ravitational acceleration .81 m/sg = g = 9 2
m ass of rocket fuel .7gΔ = m = 5 inear drag coefficient .003 kg/sb = l = 0
ass of rocket only 5.4gm r = m = 6 rag coefficient .0CD = d = 1
onstant thrust .78NTc = c = 3 uselage diameter .0254mD = f = 0
ir density .2 kg/mρ = a = 1 3
The free body diagrams for Case I, constant thrust and constant mass are shown
below. Figures 1 depicts the first segment of the flight with thrust, Figure 2 depicts the
second segment of flight without thrust.

3.
Figure 1­​ FBD Case I: Constant Thrust, Constant
Mass Segment 1: Model with Thrust
​Figure 2­​ FBD Case I: Constant Thrust, Constant
Mass Segment 2: Model After Thrust
The free body diagrams for Case II, variable thrust and constant mass are shown
below. Figures 3 depicts the first segment of the flight with thrust, Figure 4 depicts the
second segment of flight without thrust.
Figure 3­​ FBD Case II: Variable Thrust, Constant
Mass Segment 1: Model with Thrust
​Figure 4­​ FBD Case II: Variable Thrust, Constant
Mass Segment 2: Model After Thrust

4.
The free body diagrams for Case III, variable thrust and variable mass are shown
below. Figures 5 depicts the first segment of the flight with thrust, Figure 6 depicts the
second segment of flight without thrust.
Figure 5­​ FBD Case III: Variable Thrust, Variable
Mass Segment 1: Model with Thrust
​Figure 6­​ FBD Case III: Variable Thrust, Variable
Mass Segment 2: Model After Thrust
For Case I constant thrust and constant mass, the analytical and numerical models for
the rocket flight the were found in Lab #1. Figures 7, 8, and 9 show the plots of the
given acceleration data, the velocity of the rocket, and the height of the rocket. Please
see Lab #1 for a more indepth explanation on how these plots were created.

5.
Figure 7­​ Plot of the Acceleration Data of the Rocket Flight from Lab #1
Figure 8­​ Comparison Plot of the Velocity of the Rocket Flight: the Flight Under Linear Drag vs. the Flight
Under Aerodynamic Drag vs. the Experimental Data

6.
Figure 9­​ Comparison Plot of the Height of the Rocket Flight: the Flight Under Linear Drag vs. the Flight
Under Aerodynamic Drag vs. the Experimental Data
Experimental thrust data was also given, it can be seen plotted below in Figure 10.
Figure 10­​ Plot of the Experimental Data for Variable Thrust During the Rocket Flight

7.
The main goal of the lab is to develop and program a numerical and an analytical
algorithm to calculate the rocket’s velocity and height as functions of time during the
flight for all three cases and then compare. In addition, to show step size independency
for case 3 by comparing a step size of 0.05 seconds with a step size of 0.01 seconds.
Simplifying Assumptions:
The following assumptions were made:
● Horizontal displacement is negligible. The rocket experiences vertical
displacement only because the thrust acts upon the rocket solely in the vertical
direction.
● F​d​ is proportional to at “low” velocity. At “low” speeds, fluid resistance is linear v
with velocity. The proportion comes from the cohesion of the layers of the fluid.
● F​D​ is proportional to at “high” velocity. At “high” speeds, fluid resistance isv2
better resembled with a quadratic relationship with velocity. This relationship is
related to the momentum transfer between the moving object and the fluid
through which it travels.
● C​D​, b, g and are constants. The coefficients for aerodynamic and linear dragρ
are based on the shape of the rocket, the rocket does not change shape during
the duration of the model. The possible variance in the values for g and due toρ
location or weather conditions are negligible, especially if the model flights are
occurring in the same location.
● In Case I, thrust and mass are modeled as constants. This allows for the
calculations to be simplified, meaning an answer can be calculated within a tight
time constraint, if there is one presented by a customer.
● In Case II, thrust is variable and mass is modeled as a constant. This allows for
the calculations to be more accurate than Case I, since we are lying less. The
case is still partially simplified, meaning an answer can be calculated within a
time constraint, if there is one presented.
● In Case III, thrust and mass are variable, as they would be in a true experiment.
This allows for the calculations to be more accurate than Case II, since we are
lying even less.
Pre­Analysis:
Cases II and III use variable thrust, which is modeled after the experimental thrust data.
In order to allow the integration step size to be independent from the thrust data that
was given, the data was interpolated. An eighth order polynomial curve fit was found for
the thrust data using Matlab’s basic fitting tool. The equation for the line is as follows:

8.
− .37 )x 6.6202 )x − .3765 )x 65061)x 48271)x − 2321)xy = ( 3 × 105 8 + ( × 105 7 + ( 4 × 105 6 + ( 5 + ( 4 + ( 2 3
3039.4)x − 3.091)x 1.0471)+ ( 2 + ( 7 + (
Figure 11 below shows the experimental thrust data overlaid with the eight order
polynomial curve fit.
Figure 11­​ Plot of the Experimental Data for Variable Thrust During the Rocket Flight and the Eighth Order
Polynomial Best Fit Curve
Imputing time in for x to the eighth order best fit equation yields the equation for variable
thrust. This equation can be seen below.
(t) − .37 )t 6.6202 )t − .3765 )t 65061)t 48271)t − 2321)tT = ( 3 × 105 8
+ ( × 105 7
+ ( 4 × 105 6
+ ( 5
+ ( 4
+ ( 2
3039.4)t − 3.091)t 1.0471)+ ( + ( 7 + (
Case III considers variable mass along with variable thrust. The mass burn­off is
proportional to the net impulse delivered by the fuel at the current point. In other words,
the percentage of mass burned off at time t is equal to the percent of area under the
thrust­time curve from takeoff to the current time t, relative to the total impulse delivered
by all the fuel.

9.
The mass burn­off is proportional to the net impulse of the thrust through the use of a
linear momentum balance obtained from Tsiolkovsky's Rocket equation. The initial
momentum of the rocket before any fuel is consumed ,p​1​ , can be shown by the
following equation where p is momentum, Δm is the mass of the fuel, m​r​ is the mass of
the rocket without fuel, and v is the velocity of the rocket. The free body diagram for this
equation is state 1 of figure 12.
Figure 12​ ­ Linear momentum balance from Tsiolkovsky’s Rocket Equations
1 vp = m
1 m m)vp = ( r + Δ
The momentum of the rocket after thrust is initiated and mass is being burned off ,p​2​ ,
can be shown by the equation below where represents the velocity of the fuel asυ e
seen by someone on the rocket and m represents the mass of the fuel burned. TheΔ
free body diagram for this equation is state 2 of figure 12.
2 (v v) mp = mr + Δ + Δ * v e
2 (v v) m vp = mr + Δ + Δ * ( − υ )e

10. Next we can use Newton’s second law to relate the thrust to the change in momentum.
By taking the limit of this relationship as Δt approaches 0, a function for Thrust relating
the change in mass with the change in time and the relative velocity of the fuel from the
rocket can be reached. The equation and its derivation are shown below.
0 = ∑
F = dt
dp
= lim
Δt→0 Δt
Δp
= lim
Δt→0 Δt
p2−p1
lim
Δt→0 Δt
m (v+Δv) + Δm(v−υ ) − (m +Δm)v r
e
r
υ lim
Δt→0
= Δt
m Δvr* − Δt
Δm e
υ = mr dt
dv
− dt
dm e
= 0
υ dt
m dv* = dt
dm e
Recall that ​thus(t)dt
m dv* = F
(t) υ F = dt
dm e
The thrust of the rocket is the force applied to the rocket
so the equation can be rewritten in its final form as
(t) υ T = dt
dm e
Finding the indefinite integral of this new T(t) equation we get the following.
(t) t υ t m∫
T * d = ∫
dt
dm e
* d = υ e
∫
d
Finally, taking the ratio of the indefinite integral it itself with the numerator evaluated
from the initial conditions to at a specific point, t​#​
, and the denominator evaluated from
the initial conditions to the end of the thrust data the following proportion is reached.
(t) dt ∫
t#
0
T *
(t) dt∫
tfinal
0
T *
=
v me ∫
m(t#)
mfuel
d
v me ∫
0
mfuel
d
= v me fuel total
v m (t#)e fuel burned
= mfuel total
m (t#)fuel burned
This proportion shows that the mass burn­off is proportional to the net impulse of the
thrust.
Variable mass is represented by the equation:
(t) (1 (t))m = massfuel,total − Φ
where was given as 5.7g and is the percent mass burned. The percentmassfuel,total (t)Φ
mass burned is found using the proportion above.

11.
(t) Mass BurnedΦ = % =
(t)dt=v m=v m∫
tfinal
0
T e ∫
0
mfuel
d e fuel total
(t)dt=v m=v m (t )∫
t*
0
T e ∫
m(t )*
mfuel
d e fuel burned *
= mfuel total
m (t )fuel burned *
where is the mass of the fuel burned at any given time , and is(t )mfuel burned * t * mfuel total
the mass of the fuel burned over the entire duration of thrust. To find the(t )mfuel burned *
integral of the equation for variable thrust was taken. The resulting equation is seen
below:
(t) − .3744 )t 0.827525 )t − .62521 )t 10843.5)tmfuel burned = ( 0 × 105 9
+ ( × 105 8
+ ( 0 × 105 7
+ ( 6
+
9654.2)t − 580.25)t 1013.13)t − 6.5455)t 1.0471)t( 5
+ ( 5 4
+ ( 3
+ ( 3 2
+ (
To find , the total time duration of thrust, 0.55 seconds, was input into themfuel total
equation for t. The result is seen below:mfuel burned
(.55) − .3744 )(.55) 0.827525 )(.55) − .62521 )(.55)mfuel burned = ( 0 × 105 9
+ ( × 105 8
+ ( 0 × 105 7
+
10843.5)(.55) 9654.2)(.55) − 580.25)(.55) 1013.13)(.55)( 6
+ ( 5
+ ( 5 4
+ ( 3
+
− 6.5455)(.55) 1.0471)(.55)( 3 2
+ (
mfuel total .1411 g= 2
The final equation for becomes:(t)Φ
(t)Φ = 2.1411 g
(−0.3744×10 )t +(0.827525×10 )t +(−0.62521×10 )t +(10843.5)t +(9654.2)t +(−5580.25)t +(1013.13)t +(−36.5455)t +(1.0471)t5 9 5 8 5 7 6 5 4 3 2
The final equation for variable mass, m(t), then becomes:
(t) 5.7)(1 ))m = ( − ( 2.1411 g
(−0.3744×10 )t +(0.827525×10 )t +(−0.62521×10 )t +(10843.5)t +(9654.2)t +(−5580.25)t +(1013.13)t +(−36.5455)t +(1.0471)t5 9 5 8 5 7 6 5 4 3 2
The graph of variable mass vs. time (using a 0.01 second time step) can be seen below in
Figure 12.

12.
Figure 13­​ Plot of the Variable Mass of the Fuel vs. Time
Mathematical Model:
For Cases II and III, force equations were found using the free body diagrams.
Referring to Figures 3 and 4 respectively, the force equations with variable thrust and
constant mass for case II were determined:
For time 0 seconds to 0.55 seconds
F (t) g (t) g vΣ h = T − m − Fd = T − m − b = mdt
dv
For time after 0.55 seconds
F − g − g vΣ h = m − Fd = m − b = mdt
dv
Referring to Figures 5 and 6 respectively, the force equations with variable thrust and
variable mass for case III were determined:
For time 0 seconds to 0.55 seconds
F (t) m m)g (t) (m m)g vΣ h = T − ( r + Δ − Fd = T − r + Δ − b = mdt
dv
For time after 0.55 seconds

13. F − g Δmg − m)g vΣ h = mr − Fd + = ( mr + Δ − b = mdt
dv
To plot the response for Case II of the rocket flight, the equation found for thrust, T(t)
(seen in the pre­analysis section), was input into a Taylor Series based on the force
equations above.The equations for both the linear and aerodynamic drag models are
seen below.
Linear drag, for time 0 seconds to 0.55 seconds
≈ v )Δt vi+1 i + ( m
T(t)
− g − m
bvi
For time after 0.55 seconds
≈ v − )Δt vi+1 i + ( g − m
bvi
Aerodynamic drag, for time 0 seconds to 0.55 seconds
≈ v )Δtvi+1 i + ( m
T(t)
− g − 2m
ρC Av ²D i
For time after 0.55 seconds
≈ v − )Δt vi+1 i + ( g − 2m
ρC Av ²D i
To plot the response for Case III of the rocket flight, the equation found for thrust, T(t)
(seen in the pre­analysis section), is input into the Taylor Series based on the force
equations above. Additionally, the mass of just the rocket, , plus the variable massmr
m(t) is input into the equation in place of the constant mass, m. The equations for both
the linear and aerodynamic drag models are seen below.
Linear drag, for time 0 seconds to 0.55 seconds:
≈ v )Δtvi+1 i + ( T(t)
m +m(t)r
− g −
bvi
m +m(t)r
For time after 0.55 seconds, m(t) = 0:
≈ v − )Δtvi+1 i + ( g − mr
bvi
Aerodynamic drag, for time 0 seconds to 0.55 seconds:
≈ v )Δtvi+1 i + ( T(t)
m +m(t)r
− g − 2mr
ρC Av ²D i
For time after 0.55 seconds, m(t) = 0:
≈ v − )Δtvi+1 i + ( g − 2mr
ρC Av ²D i
Solutions for Unknown:

14. The velocity plots for constant thrust/constant mass found in Lab #1 were overlaid with
the variable thrust/constant mass, and variable thrust/variable mass, for both linear and
aerodynamic drag cases, as well as the experimental data. These plots can be seen
below in Figure 14. These were plotted with a time step of 0.01 seconds.
Figure 14­​ Velocity Plots for Constant Thrust/Constant Mass, Variable Thrust/Constant Mass, and Variable
Thrust/Variable Mass, for Both Linear and Aerodynamic Drag Cases, as well as the Experimental Data on a
Time Step of 0.01 Seconds
The velocity plots for constant thrust/constant mass found in Lab #1 were overlaid with
just the variable thrust/constant mass, for both linear and aerodynamic drag cases, as
well as experimental data. These plots can be seen below in Figure​ ​15. These were
plotted with a time step of 0.01 seconds.

15.
Figure 15­​ Velocity Plots for Constant Thrust/Constant Mass and Variable Thrust/Constant Mass for Both
Linear and Aerodynamic Drag Cases, as well as the Experimental Data on a Time Step of 0.01 Seconds
The velocity plots for constant thrust/constant mass found in Lab #1 were overlaid with
just the variable thrust/variable mass, for both linear and aerodynamic drag cases, as
well as experimental data. These plots can be seen below in Figure 16. These were
plotted with a time step of 0.01 seconds.

16.
Figure 16­​ Velocity Plots for Constant Thrust/Constant Mass and Variable Thrust/Variable Mass for Both
Linear and Aerodynamic Drag Cases, as well as the Experimental Data on a Time Step of 0.01 Seconds
The height plots for constant thrust/constant mass found in Lab #1 were overlaid with
the variable thrust/constant mass, and variable thrust/variable mass, for both linear and
aerodynamic drag cases, as well as the experimental data. These plots can be seen
below in Figure 17. These were plotted with a time step of 0.01 seconds.

17.
Figure 17­​ Height Plots for Constant Thrust/Constant Mass, Variable Thrust/Constant Mass, and Variable
Thrust/Variable Mass, for Both Linear and Aerodynamic Drag Cases, as well as the Experimental Data on a
Time Step of 0.01 Seconds
The height plots for constant thrust/constant mass found in Lab #1 were overlaid with
just the variable thrust/constant mass, for both linear and aerodynamic drag cases, as
well as experimental data. These plots can be seen below in Figure 18. These were
plotted with a time step of 0.01 seconds.

18.
Figure 18 ­​ Height Plots for Constant Thrust/Constant Mass and Variable Thrust/Constant Mass for Both
Linear and Aerodynamic Drag Cases, as well as the Experimental Data on a Time Step of 0.01 Seconds
The height plots for constant thrust/constant mass found in Lab #1 were overlaid with
just the variable thrust/variable mass, for both linear and aerodynamic drag cases, as
well as experimental data. These plots can be seen below in Figure 19. These were
plotted with a time step of 0.01 seconds.

19.
Figure 19­​ Height Plots for Constant Thrust/Constant Mass and Variable Thrust/Variable Mass for Both
Linear and Aerodynamic Drag Cases, as well as the Experimental Data on a Time Step of 0.01 Seconds
The figures on the following pages show the difference in height and velocity at a time
step of 0.01 seconds and 0.05 seconds for Constant Thrust, Constant Mass v.s.
Variable Thrust, Constant Mass and Constant Thrust, Constant Mass v.s. Variable
Thrust, Variable Mass.

20.
Figure 20­ ​Comparison of a time step of 0.01 seconds and 0.05 seconds for the velocity of constant thrust
and mass as well as the velocity of constant mass and variable thrust for both linear and aerodynamic drag.
Figure 21­ ​Left side of the comparison of a time step of 0.01 seconds and 0.05 seconds for the velocity of
constant thrust and mass as well as the velocity of constant mass and variable thrust for both linear and
aerodynamic drag.

21.
Figure 22­ ​Middle of the comparison of a time step of 0.01 seconds and 0.05 seconds for the velocity of
constant thrust and mass as well as the velocity of constant mass and variable thrust for both linear and
aerodynamic drag.
Figure 23­ ​Right side of the comparison of a time step of 0.01 seconds and 0.05 seconds for the velocity of
constant thrust and mass as well as the velocity of constant mass and variable thrust for both linear and
aerodynamic drag.

22.
Figure 24­ ​Comparison of a time step of 0.01 seconds and 0.05 seconds for the velocity of constant thrust
and mass as well as the velocity of variable mass and variable thrust for both linear and aerodynamic drag.
Figure 25­ ​Left side of the​ ​comparison of a time step of 0.01 seconds and 0.05 seconds for the velocity of
constant thrust and mass as well as the velocity of variable mass and variable thrust for both linear and
aerodynamic drag.

23.
Figure 26­ ​Middle of the​ ​comparison of a time step of 0.01 seconds and 0.05 seconds for the velocity of
constant thrust and mass as well as the velocity of variable mass and variable thrust for both linear and
aerodynamic drag.
Figure 27­ ​Right side of the​ ​comparison of a time step of 0.01 seconds and 0.05 seconds for the velocity of
constant thrust and mass as well as the velocity of variable mass and variable thrust for both linear and
aerodynamic drag.

24.
Figure 28­ ​Comparison of a time step of 0.01 seconds and 0.05 seconds for the height of constant thrust
and mass as well as the velocity of constant mass and variable thrust for both linear and aerodynamic drag.
Figure 29­ ​Left side of the comparison of a time step of 0.01 seconds and 0.05 seconds for the height of
constant thrust and mass as well as the velocity of constant mass and variable thrust for both linear and
aerodynamic drag.

25.
Figure 30­ ​Right side of the comparison of a time step of 0.01 seconds and 0.05 seconds for the height of
constant thrust and mass as well as the velocity of constant mass and variable thrust for both linear and
aerodynamic drag.
Figure 31­ ​Comparison of a time step of 0.01 seconds and 0.05 seconds for the height of constant thrust
and mass as well as the velocity of variable mass and variable thrust for both linear and aerodynamic drag.

26.
Figure 32­​ Left side of the​ ​comparison of a time step of 0.01 seconds and 0.05 seconds for the height of
constant thrust and mass as well as the velocity of variable mass and variable thrust for both linear and
aerodynamic drag.
Figure 33­​ Reft side of the​ ​Comparison of a time step of 0.01 seconds and 0.05 seconds for the height of
constant thrust and mass as well as the velocity of variable mass and variable thrust for both linear and
aerodynamic drag.

27. The effects from changing the time step between 0.01 seconds and 0.05 seconds are
shown on figures 20 through figure 33. The largest change in the results from changing
the time step can be seen in figures 22 and 26. In figure 22, the max velocity is found to
be 24.25 m/s for a time step of 0.05 seconds and 24 m/s for a time step of 0.01
seconds. The percent difference is 1.04% which is under 5%, the max value that we
have decided to accept error. In figure 26, the max velocity is found to be 22.65 m/s at a
time step of 0.01 seconds and 22.9 m/s at a time step of 0.05. The percent difference is
1.09%. Again, this value is under the acceptable value of 5%, the max value at which
we have decided to accept error. All other variations in the results due to the time step
change have been determined smaller that the two just discussed by inspection of the
figures.
Post Processing:
In the first portion of the rocket lab, it was assumed that the thrust of the rocket could be
approximated to as a constant value of 3.78 N for a time period of 0.55 seconds. While
this approximation was deemed to be appropriate for the lab, it is important to
understand why this approximation is being made and where the value of 3.78 N comes
from.
The equation used to determine the thrust of “particle” with a mass of ​m ​is given by
Beer (814) to be:
m )F = d
dt • ( • v
In this equation it is assumed that thrust is the force (F). Since the velocity is not
constant the equation that will be used to find the thrust will be:
T = Δt
m•Δv
In order to calculate the thrust as a constant, the thrust must be calculated using the
above equation at multiple points, and averaging the results over the time period. The
thrust was calculated approximately every 0.1 seconds. The results are shown in Table
1.
Time (s) Velocity (m/s)
0.1 2.78
0.2 13.06
0.3 18.79
0.4 20.96

28. 0.5 24.30
Table 1 ­​ Time and Velocity from the velocity graph of aerodynamics drag with variable thrust and constant
mass
The thrust values calculated between each of these points are shown in Table 2
Thrust (N)
7.3119
4.07403
1.5287
2.3747
Table 2​ ­ Thrust values calculated between points given in Table 1
By averaging the values of thrust given in Table 2, it is determined that the average
thrust of the rocket is 3.822 N. While this value is not exactly the same as the estimated
constant thrust value used in lab 1, it is within about one percent error. If more values
along the plotted lines were taken and used in the above calculations, the value would
get very close to the estimated value of 3.78 N.
Validation and Interpretation of Results:
Shown in the graphs below is the comparison between the velocity of the rocket in a few
different scenarios that were applied in this lab. The very top line shows the velocity of
the rocket when acted upon by a linear drag, when the rocket has a constant thrust and
constant mass. This the velocity graph determined in the first portion of the lab. The
difference between these velocities are shown at three separate times.

29.
Figure 34­​ Comparison of the velocity of the rocket with constant and variable thrust and mass at peak
values on a time step of 0.01 seconds

30.
Figure 35­​ Comparison of the velocity of the rocket with constant and variable thrust and mass around 0.35
seconds on a time step of 0.01 seconds

31.
Figure 36 ­​ Comparison of the velocity of the rocket with constant and variable thrust around 1 second on a
time step of 0.01 seconds
Shown in the graphs below is the comparison between the height of the rocket in a few
different scenarios that were applied in this lab. The very top line shows the height of
the rocket when acted upon by a linear drag, when the rocket has a constant thrust and
constant mass. This the height graph determined in the first portion of the lab. The
difference between these heights are shown at two separate times.

32.
Figure 37­​ Comparison of the height of the rocket with constant and variable thrust around 1.5 seconds on
a time step of 0.01 seconds
Figure 38­​ Comparison of the velocity with constant and variable thrust and mass at peak values
on a time step of 0.01 seconds

33.
Appendix:
%<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3
<3<3<3<3
% Main File : ME300LAB4.m
% Source File: N/A
% Description: This script revisits the rocket problem from week one. It
% models and compares:
% 1.)Constant thrust/constant mass model
% 2.)Variable thrust/constant mass model
% 3.)Variable thrust/variable mass model
% Author : Karly Kasten
% Date : 10/21/2015
%<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3
<3<3<3<3
clear,clc
%MODELS
%Given
m = 0.0711; %rocket mass [kg]
g = 9.81; %gravitational acceleration [m/s^2]
T = 3.78; %thrust, for 55 sec [N]
b = 0.003; %linear drag coefficient [kg/s]
p = 1.2; %air density [kg/m^3]
CD = 1.0; %drag coefficient []
D = 0.0254; %fuselage diameter [m]
r = D/2; %fuselage redius [m]
A = pi()*(r^2); %Crossectional area [m^2]
%Initial Conditions
i = 1; %Counter
v(i) = 0; %Initial velocity 0m/s (Model A) ­ Constant T/M
V(i) = 0; %Initial velocity 0m/s (Model B) ­ Constant T/M
v1(i) = 0; %Initial velocity 0m/s (Model A) ­ Var. T/Cons. M
V1(i) = 0; %Initial velocity 0m/s (Model B) ­ Var. T/Cons. M
v2(i) = 0; %Initial velocity 0m/s (Model A) ­ Var. T/M
V2(i) = 0; %Initial velocity 0m/s (Model B) ­ Var. T/M
h(i) = 0; %Initial height 0m (Model A) ­ Constant T/M
H(i) = 0; %Initial height 0m (Model B ­ Constant T/M
h1(i) = 0; %Initial height 0m (Model A) ­ Var. T/Cons. M

34. H1(i) = 0; %Initial height 0m (Model B) ­ Var. T/Cons. M
h2(i) = 0; %Initial height 0m (Model A) ­ Var. T/M
H2(i) = 0; %Initial height 0m (Model B) ­ Var. T/M
dt = 0.01; %YOU choose delta t [sec]
et = 3.00; %End time [sec]
t = 0:dt:et; %Time range [sec]
%8th Order Polynomial Curve Fit for Thrust
%Coefficients:
p1 = ­3.37e+05;
p2 = 6.6202e+05;
p3 = ­4.3765e+05;
p4 = 65061;
p5 = 48271;
p6 = ­22321;
p7 = 3039.4;
p8 = ­73.091;
p9 = 1.0471;
%Coefficients of integral:
c1=­0.3744e+05;
c2=0.827525e+05;
c3=­0.62521e+05;
c4=10843.5;
c5=9654.2;
c6=­5580.25;
c7=1013.13;
c8=­36.5455;
c9=1.0471;
%Curve Fit
x = 0:.002:.576;
y = p1*x.^8 + p2*x.^7 + p3*x.^6 + p4*x.^5 + p5*x.^4 + p6*x.^3 + p7*x.^2 + p8*x + p9;
%Mass of fuel burned as a function of time
Phi=zeros(size(t)); %Prealocate memory for loop
mass=zeros(size(t)); %Prealocate memory for loop
tf=.55; %time final NOTE: this number will be rounded for timesteps smaller than 0.001
TF=tf*(1/dt);
TotalTime = et/dt;

35. Mass_Total_Fuel = c1*tf^9 + c2*tf^8 + c3*tf^7 + c4*tf^6 + c5*tf^5 + c6*tf^4 + c7*tf^3 +
c8*tf^2 + c9*tf^1; %calculates mass of the fuel burned
for i = 1:1:TF;
Phi(i) = (c1*t(i)^9 + c2*t(i)^8 + c3*t(i)^7 + c4*t(i)^6 + c5*t(i)^5 + c6*t(i)^4 + c7*t(i)^3 +
c8*t(i)^2 + c9*t(i)^1)/Mass_Total_Fuel; %Integral of Thrust at t* over total area under
thrust curve
mass(i) = 5.7*(1­Phi(i));
end
%Constant Thrust/Mass
%Model A: Linear drag, Fd = b*v %Model B: Aerdynamic drag, FD = (1/2)*p*CD*A*V^2
%1st Segment of Asccent: t<0.55 sec, T = 3.78N
for i = 1:1:(.55/dt)
v(i+1) = v(i) + (((T/m)­g­((b*v(i))/m))*(dt)); %Diff. eq.(A) (w/ T) Velocity
h(i+1) = h(i) + (((v(i)+v(i+1))/2)*dt); %Find height
V(i+1) = V(i) + (((T/m)­g­((1/2)*p*CD*A*(V(i)^2))/m)*(dt)); %Diff. eq.(B) (W/ T) Velocity
H(i+1) = H(i) + (((V(i)+V(i+1))/2)*dt); %Find height
end
%2nd Segment of Asccent: t>0.55 sec, T = 0N
for i = (.55/dt)+1:1:TotalTime
v(i+1) = v(i) + ((­g­((b*v(i))/m))*(dt)); %Diff. eq.(A) (w/o T) Velocity
h(i+1) = h(i) + (((v(i)+v(i+1))/2)*dt); %Find height
V(i+1) = V(i) + ((­g­((1/2)*p*CD*A*(V(i)^2))/m)*(dt)); %Diff. eq.(B) (w/o T) Velocity
H(i+1) = H(i) + (((V(i)+V(i+1))/2)*dt); %Find height
end
%Variable Thrust, Constant Mass
%Model A: Linear drag, Fd = b*v %Model B: Aerdynamic drag, FD = (1/2)*p*CD*A*V^2
for i = 1:1:TF
yt = p1*t(i).^8 + p2*t(i).^7 + p3*t(i).^6 + p4*t(i).^5 + p5*t(i).^4 + p6*t(i).^3 + p7*t(i).^2 +
p8*t(i) + p9;
v1(i+1) = v1(i) + (((yt/m)­g­((b*v1(i))/m))*(dt));
h1(i+1) = h1(i) + (((v1(i)+v1(i+1))/2)*dt);
V1(i+1) = V1(i) + (((yt/m)­g­((1/2)*p*CD*A*(V1(i)^2))/m)*(dt));
H1(i+1) = H1(i) + (((V1(i)+V1(i+1))/2)*dt);
end
%2nd Segment of Asccent: t>0.55 sec, T = 0N
for i = round(TF)+1:1:TotalTime

36. v1(i+1) = v1(i) + ((­g­((b*v1(i))/m))*(dt));
h1(i+1) = h1(i) + (((v1(i)+v1(i+1))/2)*dt);
V1(i+1) = V1(i) + ((­g­((1/2)*p*CD*A*(V1(i)^2))/m)*(dt));
H1(i+1) = H1(i) + (((V1(i)+V1(i+1))/2)*dt);
end
%Variable Thrust/Mass
%Model A: Linear drag, Fd = b*v %Model B: Aerdynamic drag, FD = (1/2)*p*CD*A*V^2
for i = 1:1:TF
mr = m+(mass(i)/1000);
mR(i) = mr;
yt = p1*t(i)^8 + p2*t(i)^7 + p3*t(i)^6 + p4*t(i)^5 + p5*t(i)^4 + p6*t(i)^3 + p7*t(i)^2 +
p8*t(i) + p9;
v2(i+1) = v2(i) + (((yt/mr)­g­((b*v2(i))/mr))*(dt));
h2(i+1) = h2(i) + (((v2(i)+v2(i+1))/2)*dt);
V2(i+1) = V2(i) + (((yt/mr)­g­((1/2)*p*CD*A*(V2(i)^2))/mr)*(dt));
H2(i+1) = H2(i) + (((V2(i)+V2(i+1))/2)*dt);
end
%2nd Segment of Asccent: t>0.55 sec, T = 0N
mr=m;
for i = round(TF)+1:1:TotalTime
v2(i+1) = v2(i) + ((­g­((b*v2(i))/mr))*(dt));
h2(i+1) = h2(i) + (((v2(i)+v2(i+1))/2)*dt);
V2(i+1) = V2(i) + ((­g­((1/2)*p*CD*A*(V2(i)^2))/mr)*(dt));
H2(i+1) = H2(i) + (((V2(i)+V2(i+1))/2)*dt);
end
%USING EXPERIMENTAL DATA
%Import Excel Acceleration Data
timea = xlsread('Lab1','A2:A3362'); %Import time data
accelG = xlsread('Lab1','B2:B3362'); %Import accereration data [Gs]
accel = (accelG­1)*9.81; %Convert acceleration data to m/s^2
%Import Excel Variable Thrust Data
timet = xlsread('Thrust_Data_vs_Acceleration_2014','A1:A289'); %Import thrust time
data
thrustlbf = xlsread('Thrust_Data_vs_Acceleration_2014','B1:B289'); %Import thrust data
[lbsF]
thrust = 4.44822*(thrustlbf); %Convert thrust data to N

37.
%Obtain Experimental Velociy Data w/ Constant Thrust/Mass
j = 1; %Counter
Ve(j) = 0; %Intiial velocity 0m/s
for j = 1:1:3360
Ve(j+1) = Ve(j) + ((accel(j)+accel(j+1))/2)*(timea(j+1)­timea(j));
end
%Obtain Experimental Position Data w/ Constant Thrust/Mass
k = 1; %Counter
He(k) = 0; %Initial position 0m
for k = 1:1:3360
He(k+1) = He(k) + ((Ve(k)+Ve(k+1))/2)*(timea(k+1)­timea(k));
end
%ALL THE PLOTS
%Plot Acceleration
figure(1) %Opens New Figure
%hold on
plot(timea,accel,'m­­')%Plot
xlabel('Time [s]') %x label
ylabel('Acceleration [m/s^2]') %y label
grid on %Turn on/off grid
%Plot Variable Thrust & Best Fit
figure(2) %Opens New Figure
%hold on
plot(timet,thrust,'b:',x,y,'m­­')%Plot
xlabel('Time [s]') %x label
ylabel('Thrust [N]') %y label
legend('Experimental Data, Variable Thrust','8th Order Best Fit') %Legend
grid on %Turn on/off grid
%Plot Velocities w/ Constant Thrust/Mass
figure(3) %Opens new figure
plot(t,v,'g­­',t,V,'b:',timea,Ve,'m­.') %Plots functions
xlabel('Time [s]') %x label'
ylabel('Velocity [m/s]') %y label
legend('Linear Drag, Constant Thrust/Mass','Aerodynamic Drag, Constant
Thrust/Mass','Experimental Data, Constant Thrust/Mass') %Legend

38. grid on %Turn on/off grid
%Plot Heights w/ Constant Thrust/Mass
figure(4) %Opens new figure
plot(t,h,'g­­',t,H,'b:',timea,He,'m­.') %Plots functions
xlabel('Time [s]') %x label
ylabel('Height [m]') %y label
legend('Linear Drag, Constant Thrust/Mass','Aerodynamic Drag Constant
Thrust/Mass','Experimental Data, Constant Thrust/Mass') %Legend
grid on %Turn off/on grid
%Plot Velocities w/ Variable Thrust, Constant Mass
figure(5) %Opens new figure
plot(t,v1,'g­­',t,V1,'b:') %Plots functions
xlabel('Time [s]') %x label'
ylabel('Velocity [m/s]') %y label
Legend('Linear Drag, Var. Thrust/Cons. Mass','Aerodynamic Drag, Var. Thrust/Const.
Mass') %Legend
grid on %Turn on/off grid
%Plot Heights w/ Variable Thrust, Constant Mass
figure(6) %Opens new figure
plot(t,h1,'g­­',t,H1,'b:') %Plots functions
xlabel('Time [s]') %x label
ylabel('Height [m]') %y label
legend('Linear Drag, Var. Thrust/Cons. Mass','Aerodynamic Drag, Var. Thrust/Const.
Mass') %Legend
grid on %Turn off/on grid
%Plot Velocities w/ Variable Thrust/Mass
%figure(7)%Opens new figure
%plot(t,v2,'g­­',t,V2,'b:') %Plots functions
%xlabel('Time [s]') %x label'
%ylabel('Velocity [m/s]') %y label
%legend('Linear Drag, Var. Thrust/Mass','Aerodynamic Drag, Var. Thrust/Mass')
%Legend
%grid on %Turn on/off grid
%Plot Heights w/ Variable Thrust/Mass

39. %figure(8) %Opens new figure
%plot(t,h2,'g­­',t,H2,'b:') %Plots functions
%xlabel('Time [s]') %x label
%ylabel('Height [m]') %y label
%legend('Linear Drag, Var. Thrust/Mass','Aerodynamic Drag, Var. Thrust/Mass')
%Legend
%grid on %Turn off/on grid
%COMPARE PLOTS
%Velocities ­ Constant T/M vs. Var. T/ Const. M
figure(9) %Opens new figure
%hold on
plot(t,v,'g­­',t,V,'b:',timea,Ve,'m­.',t,v1,'r­­',t,V1,'y:',t,v2,'c­­',t,V2,'­.') %Plots functions
xlabel('Time [s]') %x label'
ylabel('Velocity [m/s]') %y label
legend('Linear Drag, Constant Thrust/Mass','Aerodynamic Drag, Constant
Thrust/Mass','Experimental Data, Constant Thrust/Mass','Linear Drag, Var. Thrust/Cons.
Mass','Aerodynamic Drag, Var. Thrust/Const. Mass','Linear Drag, Var. Thrust/Var.
Mass','Aerodynamic Drag, Var. Thrust/Var. Mass') %Legend
grid on %Turn on/off grid
%Heights ­ Constant T/M vs. Var. T/ Const. M
figure(10) %Opens new figure
%hold on
plot(t,h,'g:',t,H,'b­­',timea,He,'m­.',t,h1,'r­­',t,H1,'y:',t,h2,'c­­',t,H2,'­.') %Plots functions
xlabel('Time [s]') %x label
ylabel('Height [m]') %y label
legend('Linear Drag, Constant Thrust/Mass','Aerodynamic Drag Constant
Thrust/Mass','Experimental Data, Constant Thrust/Mass','Linear Drag, Var. Thrust/Cons.
Mass','Aerodynamic Drag, Var. Thrust/Const. Mass','Linear Drag, Var. Thrust/Var.
Mass','Aerodynamic Drag, Var. Thrust/Var. Mass') %Legend
grid on %Turn off/on grid
%Mass of the fuel burned per unit time
figure(11)
%hold on
plot(t,mass,'­­')
axis([0 0.6 0 6])
xlabel('Time [s]'),ylabel('Mass of Fuel [g]')

40. legend('Fuel burned')
grid on
%All the comparisons for lab
%Velocity Constant thrust/mass & Var. thrust/cons. mass
figure(12)
hold on
plot(t,v,'g­­',t,V,'b:',t,v1,'r­­',t,V1,'y:') %Plots functions
xlabel('Time [s]') %x label'
ylabel('Velocity [m/s]') %y label
legend('Linear Drag, Constant Thrust/Mass','Aerodynamic Drag, Constant
Thrust/Mass','Linear Drag, Var. Thrust/Cons. Mass','Aerodynamic Drag, Var.
Thrust/Const. Mass') %Legend
grid on
%Velocity Constant thrust/mass & Var. thrust/Var. mass
figure(13)
hold on
plot(t,v,'g­­',t,V,'b:',t,v2,'c­­',t,V2,'­.') %Plots functions
xlabel('Time [s]') %x label'
ylabel('Velocity [m/s]') %y label
legend('Linear Drag, Constant Thrust/Mass','Aerodynamic Drag, Constant
Thrust/Mass','Linear Drag, Var. Thrust/Var. Mass','Aerodynamic Drag, Var. Thrust/Var.
Mass') %Legend
grid on
%Height Constant thrust/mass & Var. thrust/cons. mass
figure(14)
hold on
plot(t,h,'g­­',t,H,'b:',t,h1,'r­­',t,H1,'y:') %Plots functions
xlabel('Time [s]') %x label'
ylabel('Height [m]') %y label
legend('Linear Drag, Constant Thrust/Mass','Aerodynamic Drag, Constant
Thrust/Mass','Linear Drag, Var. Thrust/Cons. Mass','Aerodynamic Drag, Var.
Thrust/Const. Mass') %Legend
grid on
%Height Constant thrust/mass & Var. thrust/Var. mass
figure(15)
hold on
plot(t,h,'g­­',t,H,'b:',t,h2,'c­­',t,H2,'­.') %Plots functions

41. xlabel('Time [s]') %x label'
ylabel('Height [m]') %y label
legend('Linear Drag, Constant Thrust/Mass','Aerodynamic Drag, Constant
Thrust/Mass','Linear Drag, Var. Thrust/Var. Mass','Aerodynamic Drag, Var. Thrust/Var.
Mass') %Legend
grid on