THE APPLICATION OF NEWTONIAN PHYSICS IN SIMULATED AND
EXPERIMENTAL MODEL ROCKETRY
Tarung Bhimnathwala, Colette Bilynsky, Cara Giovanetti, David Hua, Katie Liu, Anand
Nanduri, Max Nezdyur, Vineeth Puli, Grace Stridick, Farhan Toddywala, Meghan Wilmott,
Jason Xu
Advisor: Robert Murawski
Assistant: Sam Zorn
ABSTRACT
Model rocketry provides amateurs a glimpse into the diversity and possibility of the field
of rocket science. Throughout this project, the research team delved into the world of model
rocketry, and discovered how to connect calculations, simulations, and experimental flights of
model rockets. To start, the team derived the Tsiolkovsky rocket equation to understand the
relationship between the rockets’ decreasing masses and their changes in velocities. Each team
member constructed a rocket using one of the three models and then utilized a flight simulation
program called OpenRocket to investigate the predicted flight trajectory for each unique rocket
(1). This data was compared to experimental flights conducted by the team, during which the
effects of modifications of variables such as rocket type, payload mass, and engine type were
investigated. In the first study, different models of rockets and their apogees were compared.
The different physical characteristics of the rocket models were also analyzed. In the second
trial, the experimental thrust curves were determined to corroborate with the thrust curves
published by the company, Estes. In the final study, the team analyzed the effects of increasing
payload upon a rocket’s apogee. Throughout the trials, the team concluded that the simulations
from OpenRocket generally agreed with the experimental flights, despite difficulty measuring
certain variables in the field. Through the duration of this project, the group was able to
understand the physics behind model rockets and the factors that contribute to a successful
launch.
INTRODUCTION AND HISTORY
Rocketry has a vast and long history. Most structures and
methods were not discovered with the intention of space
exploration, but rather led to the development of current rockets
and spacecrafts. The effective birth of rocketing methods came
about with Hero of Alexandria’s development of the aeolipile,
which somewhat serendipitously illustrated Newton’s third law by
utilizing steam as a means of propulsion. The aeolipile is a
steam-powered engine consisting of a sphere mounted over a
kettle with two bent tubes leading out, out of which steam could
be directed (Fig. 1). As steam rushes out, the apparatus spins (3).
Although far from a true rocket, this invention laid crucial
foundation for the principles of modern rockets, as it
demonstrated how thrust can serve as a powerful tool when
attempting to move an object, especially against gravity.
[1-1]
Figure 1: The Hero of Alexandria’s
aeolipile was one of the first
examples of using steam as
propulsion. This is showed how
thrust can be a powerful force to
move an object. (2)
The first “true rockets” did not appear until 1232. During this 13-century
interim, the Chinese had developed and nearly perfected gunpowder. This new,
powerful invention replaced steam as a means of propulsion and allowed objects to
be moved greater distances at greater speeds (4). Many Chinese developed their
technology to the point where they could attach small canisters filled with
gunpowder to arrows. These weapons would effectively propel themselves into
enemy ranks (5).
Improving upon previous technology, in 1591, Johann Schmidlap
attempted to design fireworks that could reach greater heights than ever before. He
developed what he described as “step rockets”, now known as staging (6). These
Figure 2:
fireworks utilized not one, but two rockets for propulsion (Fig. 2). As the first
Schmidlap’s
Rocket was the
stage burned out, it would fall back to the earth, making the second stage more
first “step
efficient and capable of reaching higher altitudes (6).
rocket” or a
By 1687, rocketry had truly earned the distinction of a “science”, as this
rocket with
staging. It used
technology was finally established to have capitalized on Newton’s recently
two rockets for
formulated third law of motion (3). With the publication of Principia, many
propulsion,
were able to visualize the fundamental concepts behind this technology that had
allowing it to
reach very high
been in use for centuries, spurring further understanding and further
altitudes. (7)
development.
As rocket technology lurched forward, it became apparent that a single,
unifying mathematical expression, based on a rocket with changing mass due to
exhaustion of fuel, was required in order to further advance the science. Konstantin
Tsiolkovsky, now known as the father of modern aeronautics, derived his famous rocket
equation while exploring the viability of space exploration in 1898. Tsiolkovsky’s equation
accounted for the mass of the fuel expelled in order to generate thrust, while predicting the final
velocity of the rocket as well (8).
Prior to 1919, the highest altitudes known to man
were reached by way of lighter-than-air balloons. American
scientist Robert H. Goddard shattered that precedent upon
the publication of his pamphlet “A Method of Reaching
Extreme Altitudes”; a feat achieved with rockets (3).
Goddard did further research with solid fuel rockets, and,
more importantly, with liquid fuel; a task never successfully
accomplished before his experiments in 1926 (Fig. 3).
Reaching an apogee of 12.5 meters and landing a mere 56
Figure 3: Goddard’s Rocket utilized liquid
meters from the launch pad, Goddard’s oxygen and
fuel, specifically oxygen and gasoline. His
gasoline fueled rocket ushered in a new era of progress in
research with liquid fuel and parachute
rocketry (3). He continued to
systems made him the father of modern
research his liquid fueled rockets, as rocketry. (9)
well as parachute systems and other
such developments, thereby earning his title as the father of modern
rocketry.
On October 4th, 1957, the Soviet Union amazed the entire globe.
Sputnik I, the first ever Earth-orbiting artificial satellite, was launched
into outer space (Fig. 4). Not only was the launch of this satellite the
Figure 4: Sputnik was the
first ever Earth-orbiting
catalyst for the space race between the United States and the Soviet
satellite. It was launched by
the Soviet Union sparking
the Space Race. (10)
[1-2]
Union, but it was also the precursor to a space program in the U.S (3). The space programs in
nations around the world, particularly in the United States and the former Soviet Union,
blossomed. In 1969, the United States rose to the challenge previously presented by the Soviets
and put a man on the moon (11). The first component of the International Space Station, an
unprecedented cooperative research platform, was launched in 1998 (12). These incredible feats
could not have been accomplished without the assistance of the major developments in rocketry
during the previous few centuries.
THEORY
The foundation of rocket science lies in Newton’s three laws of motion, which govern the
fundamentals of rocket flight. Newton’s first law states that a system travels at constant velocity
unless acted upon by a net external force. Newton’s second law demonstrates that the net
external force is equal to the derivative of the system’s momentum, the product of its mass and
velocity, with respect to time. According to Newton’s third law, every force occurs together
with another force of equal magnitude in the opposite direction. Since force is equal to the
change in momentum divided by the time differential, the momentum remains constant. Thus,
Newton’s second and third laws together imply that momentum is conserved; a system with only
internal forces will experience no net change in momentum.
The Tsiolkovsky equation is derived from these fundamental principles by considering
the rocket body and its exhaust as a single system with initial velocity v and initial mass m.
Internal stresses and forces in the rocket and molecular interactions in the exhaust can be
neglected, as they do not affect the momentum of the system. By design, the rocket expels hot
exhaust from its base to produce thrust; most rocket engines accomplish this task by utilizing a
highly exothermic reaction that releases gases at high speeds. The engine, as a result of its
orientation within the model rocket, then directs these gases through the bottom of the rocket.
Although the vast majority of rocket motors use a multistage design with variable thrust, the
classic Tsiolkovsky equation assumes that the rate of gas expulsion, or thrust, is constant.
Using the momentum-impulse equation of the rocket, the value pt represents the
momentum of the system with mass m and velocity v at a time t. In a differential time slice dt,
the exhaust has a momentum of dpe, which is the product of its respective mass dme and velocity
ve. Since momentum is conserved, the rocket accelerates in the opposite direction, picking up a
velocity dv. It will also lose a mass dm in the form of exhaust.
In accordance with the description above, the initial momentum of the system at time t is
pt = mv
(1)
After a time dt passes, the new momentum, which is affected by exhaust being expelled in the
time dt, is
pt + dt = (m - dme)(v + dv)+ dme(v +dv - ve)
(1.2)
This multiplies out to
pt + dt = mv + mdv - dmev - dmedv + dmev + dmedv - dmeve
(1.3)
[1-3]
Which, after eliminating products of differentials, simplifies to
pt + dt = mv + mdv - vedme
(1.4)
If we assume the external forces on the rocket sum to zero, the following expression can be
written and derived from the momentum-impulse equation. At any given instant, the momentum
gained by the rocket system is equal to the momentum gained by the exhaust particles in the
opposite direction. Thus, the net change in momentum is zero. The force F on any system is
equal to the change of its momentum over a corresponding change in time. Since the change in
momentum of the system is zero, the total force on the system is also zero.
Fnet =
𝑑𝑑
𝑑𝑑
=0
(2)
Cross-multiplying dt, we arrive at the product of the net force on the system and the time slice dt,
which is equal to the change in the momentum.
𝛥𝑑 =pt + dt - pt = 0
mv + mdv - vedme - mv = 0
mdv - dmeve = 0
(2.1)
Because the increase of the mass of the exhaust is equal to the decrease in the rocket’s mass, we
can replace the differential of the exhaust mass with the differential of the rocket mass.
mdv + vedm = 0
𝑑
dv = - 𝑑dm
𝑑
(2.2)
By taking the integral of both sides of the equation, we can arrive at an expression for the change
in velocity in terms of the change in mass of the rocket
∫ 𝑑𝑣 = ∫ −
𝑣𝑒
𝑚
(2.3)
Solving the integral and pulling out the constant of the negative exhaust velocity, we arrive at a
simple integral expression.
∆𝑣 = −𝑣𝑒 ∫
1
𝑑𝑚
𝑚
(2.4)
After rearranging the logarithmic expression, the negative sign goes away as the initial mass and
final mass are flipped in the equation.
𝑚
)
𝑚0
𝑚
∆𝑣 = 𝑣𝑒 ln( 0)
∆𝑣 = −𝑣𝑒 𝑙𝑛 (
𝑚
(2.5)
[1-4]
Tsiolkovsky’s equation does not account for the aerodynamic forces on the rocket. In
any real launch, four major forces act on a rocket: lift, gravity,
thrust, and drag. The drag force acts as a frictional force, which
opposes the direction of motion, and in the case of an ascending
rocket, acts downwards. Moreover, the thrust is not constant;
instead it follows a set curve with respect to time (Fig. 5). At
liftoff, engine power increases rapidly. As the rocket continues
to climb, the thrust decreases until it reaches a slow, stable thrust
power. Eventually, the rocket begins to run out of fuel and the
thrust decreases more steeply. In a model rocket, these three
steps can occur well within a single second.
Two critical points in a rocket, center of gravity and center of Figure 5: A thrust curve
demonstrating how the thrust
pressure, are defined as specific points in a rocket’s design. The
increases rapidly until it runs
center of gravity is the average location of the weight of an object,
out of fuel and then thrust
decreases.
while the center of pressure is the average location of the pressures
acting on the rocket. The force of gravity acts downwards upon the center of gravity, working
against thrust and pulling the rocket back to Earth. Lift acts at the center of pressure, and
generally provides a stabilizing force, keeping the rocket traveling in a relatively straight path.
This force acts at the center of pressure and rotates the rocket about its center of mass. If the
rocket is well-designed, the center of pressure will be below the center of mass and the lift will
stabilize the rocket in a vertical trajectory. If the center of pressure is above the center of mass,
the lift force will destabilize the rocket, causing it to veer away from the flight path and possibly
propel itself back towards Earth.
In the team’s experiments, the motor was small in relation to the rocket body, so the loss
of mass did not significantly affect the flight. However, the high launch speeds caused
significant drag on the rockets, greatly affecting the trajectory. To help model this situation
more accurately, a different equation that assumes mass and thrust to be constant while
accounting for drag, was derived.
At high speeds, drag, which is c, is proportional to the square of the velocity. It is also
proportional to the surface area of the rocket, which is 𝜌, and air density, which is A, expressed
as
(3)
Other forces on the rocket body include the thrust and gravity, which is g. Using Newton’s
second law, and assuming constant mass, the net force on the rocket body is
(3.1)
The acceleration is then
𝑑𝑣 𝑇
1 𝐶𝑑 𝜌𝐴𝑣 2
= −𝑔−
𝑑𝑡 𝑚
2 𝑚
[1-5]
(3.2)
This can be simplified by defining two constants C1 and C2 as
𝐶1 =
𝐶2 =
𝑇
−𝑔
𝑚
1 𝐶𝑑 𝜌𝐴
2 𝑚
(3.3)
(3.4)
changing the original equation as
𝑑𝑣
= 𝐶1 − 𝐶2 𝑣 2
𝑑𝑡
(3.5)
After separating variables, this becomes
𝑑𝑣
= 𝑑𝑡
𝐶1 − 𝐶2 𝑣 2
(3.6)
This equation can be integrated to obtain
∫
𝑑𝑣
= ∫ 𝑑𝑡
𝐶1 − 𝐶2 𝑣 2
1
𝐶2
−1
𝑡𝑎𝑛ℎ (√ 𝑣) = ∆𝑡
𝐶1
√𝐶1 𝐶2
(3.7)
And, finally, solving for v yields
𝐶1
𝑣 = √ 𝑡𝑎𝑛ℎ(√𝐶1 𝐶2 ∆𝑡)
𝐶2
(3.8)
The above equation can be used in analysis of a model rocket’s flight to determine velocity at
any given time. Since velocity is equivalent to the derivative of the position value in respect to
time, we can take the integral of both sides through separation of variables
𝑣 =
𝑑𝑦
𝐶1
= √ tanh(√𝐶1 𝐶2 ∆𝑡
𝑑𝑡
𝐶2
𝐶1
∫ 𝑑𝑦 = √ ∫ 𝑡𝑎𝑛ℎ (√𝐶1 𝐶2 𝑡)𝑑𝑡
𝐶2
[1-6]
(3.9)
Therefore, the position formula can be expressed through integration as
𝐶1 𝑙𝑛|𝑐𝑜𝑠ℎ(√𝐶1 𝐶2 𝑡)|
𝑦= √
𝐶2
√𝐶1 𝐶2
𝑦=
𝑙𝑛|𝑐𝑜𝑠ℎ(√𝐶1 𝐶2 𝑡)|
𝐶2
(3.10)
This equation can be used to determine the vertical position of a model rocket at any given time
during the thrust period
Because of the inertial property of the rocket, there is still a duration where the rocket is still
rising. The team derived an equation to model the motion of the rocket at a given time after the
thrust engine is exhausted. The only forces acting on the rocket at that moment would be the
gravitational force and the drag force, which can be expressed through Newton’s Second Law.
𝑑𝑣
= −𝑚𝑔 − (𝐶𝑑 𝜌𝐴)𝑣 2
𝑑𝑡
(𝐶𝑑 𝜌𝐴) = 𝐶 (𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑚
(4)
Through separation of variables and integration, an expression of time as a function of velocity
can be created.
1
𝑑𝑣
− ∫
= ∆𝑡
𝑔 1 + 𝐶 𝑣2
𝑚𝑔
(4.1)
An equation that can express the position of the rocket is significantly more useful for
quantitative data. By integrating taking the second integral of the original equation, we find an
equation for position after the engine halts as a function of time.
𝑚 × 𝑙𝑛 |𝑐𝑜𝑠 (𝑡𝑎𝑛−1 (√
𝑦=
∆𝑡√𝐶𝑔
𝐶
𝑣𝑜 ) −
)|
𝑚𝑔
√𝑚
𝐶
(4.2)
[1-7]
EXPERIMENTAL PROCEDURE
OpenRocket
The trajectory path of the rockets were simulated on a program called OpenRocket, which
allowed us to test various parameters and determine the apogee of each model (1). This software
allowed students to predict this data prior to field testing, providing a guideline for expected
results for each launch. The team was able to reconstruct their rockets from parts loaded in the
Estes database to effectively predict the apogee of their rockets and compare the simulation to
the experimental launch.
Experimental Launch
Models have a unique flight and recovery method as compared to genuine, full scale
rockets. During field testing, prior to launch, each rocket is prepped with an engine and some
wadding beneath the recovery device. The rockets are placed on a launchpad and threaded onto
the metal rod, shooting upward in a generally straight path until the engine is almost out of
power. At this point, the engine releases one final charge, popping the nose cone off and causing
the parachute or streamer to deploy. Both recovery devices increase drag to slow the rocket
down and allow for a smoother descent back towards the ground. If all aspects of the launch go
well, the models are able to be launched multiple times, allowing for experimentation.
Figure 6: A comparison between Estes models Alpha (L),
Viking (C), and Generic E2X (R) all of which the team
constructed and tested.
Three models of Estes rockets were used for the experimental launch: Alpha, Viking, and
Generic E2X (Fig. 6). Each rocket consisted of a cardboard body tube, but most other
characteristics, including mass, number and shape of fins and cross-sectional area, were unique
to the model.
During launch, two “anglers” (researchers with altimeters) were placed on the field at a
set distance from each other, and the rocket was placed equidistant from and collinear to the
anglers (Fig. 7). For very high anticipated apogees, the anglers were separated further, as the
altimeters could only measure angles up to 70º. Using trigonometry, the apogee was determined
from an average of the two angles measured at the time the rocket parachute deployed.
[1-8]
Each of the twelve rockets assembled by the team
members was tested with different variables altered each
time, providing the team with a comprehensive study of
the effects of certain variables upon a rocket’s flight path.
Specifically, three individual tests were conducted to
determine certain information regarding the influence
different factors have upon the apogee of a model rocket.
Figure 7: Two recorders stood equidistant
from the launch pad with altimeters. The team
then averaged the angles measured with the
altimeters and used trigonometry to solve for
h.
DATA & ANALYSIS
The team conducted experiments to provide data in the following three studies:
1. Comparison of apogee in OpenRocket simulation vs. experimental launch
2. Acceleration vs Time: Comparison of thrust in Estes vs. experimental launch
3. Effect of payload on apogee
Study 1
The first study compared the apogee of the rockets with the apogee of the simulated
rockets designed in the OpenRocket computer modeling program (1). Each launch was
simulated on OpenRocket, a computer program that realistically simulates model rocket flight
patterns. Within the software, each simulated rocket was created to match the physical
characteristics of the real rocket. The simulations were run under ideal conditions (i.e. no wind
and little turbulence) and estimated the vertical heights as a function of time (Fig. 8).
Figure 8: An example graph of a rocket’s flight modeled on OpenRocket. The
program showed times of motor burnout, parachute deployment, and apogee.
[1-9]
The simulated data was compared to the experimental apogees obtained from the
launches. Additionally, the times needed for the rockets to reach apogee were recorded and
compared to the times obtained from the OpenRocket simulation. These comparisons were
shown numerically by calculating the percent error for each individual rocket’s launch. The data
from the launches demonstrates that the experimental model rockets did in fact compare
reasonably (with an average error of < 16% error) with the OpenRocket simulations.
The first trial used a set of four Alpha rockets. Alpha rockets each employed three balsa
wood fins and had an average mass of approximately 28.94 grams. Table I depicts both the
experimental apogee versus the simulated apogee and the experimental time to apogee versus the
simulated time of the Alpha rockets. Both data collections are compared to the simulation to
generate percent error.
Table I: Alpha Rocket Data
Mass of
Rocket (g)
Average
Angle (°)
Max
Height (ft)
Simulation
Height (ft)
Time (s)
Simulation
Time (s)
% Error
Height
% Error
Time
29.06
62.5
290.74
303.9
5.65
4.10
4.33
37.80
30.00
59.0
249.93
296.94
3.66
4.05
15.83
9.60
30.15
60.0
259.81
295.67
6.4
4.10
12.12
56.09
30.16
59.0
249.64
296.1
4.5
4.20
15.69
7.14
The second trial used a set of four Viking rockets.
The Viking rockets were each constructed using different
number of fins and an unique fin shape (Fig. 9), each rocket
was individually created and simulated in OpenRocket to
match its actual design. It is inconclusive if the number and
orientation of fins affects the apogee and time of flight.
Figure 9: Various numbers and shapes of fins were
On average, Viking rockets had a lower mass than the
out on the Viking rockets. Two rockets had 5 fins,
other rocket designs, with their average mass being
one had 4, and the last had 3.
approximately 17.56 grams. Table II depicts both the
experimental apogee versus the simulated apogee and the experimental time to apogee versus the
simulated time of the Viking rockets.
Table II: Viking Rocket Data
Mass of
Rocket
(g)
Average
Angle (°)
Max
Height (ft)
Simulation
Height (ft)
Time (s)
Simulation
Time (s)
% Error
Height
% Error
Time
16.10
62.5
387.66
335.00
3.88
4.30
15.72
7.45
16.29
70.0*
549.49
369.60
4.00
4.51
48.67
11.31
17.16
62.0
376.15
399.00
4.35
4.70
5.72
7.45
20.70
70.0*
412.12
295.00
4.47
4.20
39.70
6.43
[1-10]
*Because the highest measurable angle with the equipment is 70º, these launches may be invalid because the rocket may have
traveled further upward than measurable or the angle measures may have not locked properly which allowed the angle meter to
swing to the higher value.
The final trial used a set of four Generic E2X rockets. The Generic rockets each had four
plastic fins, and the models had an average mass of 39.84 grams. Table III depicts both the
experimental apogee versus the simulated apogee and the experimental time to apogee versus the
simulated time of the Generic rockets.
Table III: Generic Rocket Data
Mass of
Rocket (g)
Average
Angle (°)
Max
Height (ft)
Simulation
Height (ft)
Time (s)
Simulation
Time (s)
% Error
Height
% Error
Time
37.90
50.5
201.97
195.6
4.28
3.8
3.26
4.59
38.83
43.5
242.67
194.5
3.87
3.70
24.77
12.36
41.07
45.0
190.04
184.34
5.15
3.80
3.09
35.53
42.41
33.5
132.74
178.26
3.59
3.80
25.54
5.53
These trials display the clear difference between the Alpha, Viking, and Generic rocket
models. Despite small variations due to different people constructing each rocket, the apogees of
each model are clearly grouped in clusters (Fig. 10).
Figure 10: This graph shows data points displaying the relationship between
mass and height. The Generic, the heaviest in green, is grouped with the
lowest apogee. Alpha, slightly lighter in blue, had a higher apogee. The
Viking, the lightest in red, had the highest apogee.
[1-11]
This grouping corresponds to the masses of the rockets. The Viking rockets were the
lightest and had the highest apogees. The Alpha models, which had masses between Viking and
Generic E2X, generally had apogees higher than the Generic but lower than the Viking. Finally,
the Generic E2X rockets were heaviest, and had the lowest apogees of all of the models. These
trials suggest that there is a correlation between mass and highest apogee, a concept explored
more thoroughly in Study 3. However, the various models all used different materials and had
different fin shapes which also contributed to the variation within the apogees between models.
Tables I, II, and III depict comparisons between the experiment and simulated data of
apogee and illustrate that the model rockets do in fact fly relatively similar to the simulation.
The experiment proved that the OpenRocket software correctly models the rocket flights. It
takes into consideration different conditions and simulates the physical characteristics of the
rockets. The Alpha rockets had an average of 12.00% error for maximum apogee, the Generic
rockets had an average of 14.17% error–discounting the skewed launches which were the
statistical outliers–, and the Viking rockets had an average of 10.36% error. The study showed
the variation between the rockets as well as illustrating the generally accurate correlation
between the simulation and experimental data.
Study 2
This study was conducted using slow-motion cameras and careful frame-by-frame
analysis in order to compare the thrust calculated from the data from our model rocket launches,
and the Estes’ advertised thrust of the engines. To collect data for the rocket’s thrust phase, a
camera was used to record the rocket as it was launched from its starting position on the
ground. These videos were then used to pinpoint the rocket’s location in the y-direction in each
frame (the number of frames per second varied depending on the type of video taken; regular
videos were viewed at 50 frames per second, while slow motion videos were viewed at 240
frames per second) using the video analysis program Tracker (13). Note that the first datum for
each launch video was collected right before the rocket began to move upward. A meter stick
was placed in the ground parallel to the launch rod so that the data for the video could be
correctly calibrated. One important source of error to note was the appearance of multiple “ghost
rockets” in the slow-motion videos, i.e. multiple rockets appearing in one frame and/or not
changing location from one frame to the next. These video flaws compromise the accuracy of
the data collected from such videos using Tracker.
Data was collected for each rocket launch and resulted in varying degrees of
compatibility with the Estes data. An example of the position versus time graph for a Viking
rocket is displayed below in Figure 11:
[1-12]
A cubic regression was used with the data from each launch using Wolfram Mathematica,
thereby obtaining an equation for the vertical displacement, y, as a function of time t for each
launch (14). For this example, the vertical displacement vs. time equation was y(t)=905.613t .
We could then use this equation to determine the thrust vs. time equation for each launch as the
rocket was leaving the launchpad and compare it to the A8-3 engine thrust equation given by
Estes.
To determine the A8-3 engine thrust equation, we took Estes’ thrust versus time data and
found the point of maximum thrust, (0.226s, 9.73N) (Fig. 12). Having both this point and the
point (0s, 0N), we could calculate the slope, b, of the thrust graph, which we determined to be
43.05 N/s. We knew that the function of thrust versus time for the period during which the rocket
was leaving the launchpad should be in the form T=bt, so for the Estes data the thrust equation
was T(t)=43.05t. Note that, since the units of thrust are newtons, the equations for b are ft/s .
In order to determine the b value for our launch, we first had to convert our vertical
displacement vs. time equation from feet to meters. Our original b value for this specific rocket
launch was 905.6131 ft/s , which is equivalent to approximately 276.031 m/s . The mass of this
specific rocket without the engine was 16.29 g and the mass of the engine was 13 g, so the total
payload for this launch was 29.29 g or 0.02929 kg.
We then had to make some assumptions to simplify the equation which accounts for all
forces acting on the rocket (except for friction of the launch rod), mdvdt=T-W- . For most of this
section of the thrust vs. time curve, however, thrust (T) was much greater than weight (W), and
since the rocket is starting from a velocity of 0 m/s the force of drag (D) is negligible. This
equation can therefore be simplified to mdvdt=T, which can be rewritten as dvdt=Tm. Since
T=bt, we can rewrite this as dvdt=btmor dv=btmdt, and integrating both sides results in the
equation v=b2mt . Since v=dydt, so we can write the rearranged equation dy=b2mt dt and
integrate both sides to obtain the equation y=b6mt . We can set the leading coefficient of the
vertical displacement vs. time equation from our own launch, which in this example was 276.031
m/s , and set it equal to b6m to solve for b, which for this specific example results in
b=6(.02929)(276.031)=48.51. We can then compare this b-value to Estes’ and determine the
percent error, which for this launch was 12.68%.
3
3
3
3
D
2
2
3
3
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Figure 12: A8 Thrust graph given by Estes that was used to calculate the acceleration of datapoints. (15)
Our equations for acceleration of the rocket during the initial thrust period of the rocket is
relatively close to Estes’ expected acceleration equation for this same period (15).
Study 3
This study was conducted to
analyze the effects of payload
increases on the apogee of an Alpha
rocket. As five gram increments of
lead were added to the nose cone,
there was a clear effect on the apogee
as the maximum height generally
decreased after each launch (Table
IV). Compared to the OpenRocket
simulation in (Fig. 13), the rocket
performed similarly to how it was
expected to perform under ideal
conditions. The Alpha rocket
performed rather well with the
Figure 13: Line graph that shows relationship between mass and apogee in average percent error being 13.72%
the experimental launches and the simulation.
for the apogees and 8.02% for the
times of flight to the apogee. As the
simulation predicted, and the trials confirmed, adding more mass decreases the apogee of the
rocket (Fig 13).
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Additionally, as the payload of an Alpha rocket increased, the acceleration was shown to
decrease: as the payload increased, the time it took for the rocket to reach a common height of
12 feet with varying payloads increased as well.
Table IV: Varying Payload Data
Mass of
Rocket (g)
Average
Angles (°)
Max Height
(ft)
Simulation
Height (ft)
Time (s)
Simulation
Time (s)
% Error
Height
% Error
Time
34.41
56.0
296.51
259.2
3.72
3.8
14.4
2.10
40.1
41.0
173.85
218.9
3.44
3.8
20.06
9.28
45.61
44.5
196.54
185.5
3.56
3.7
5.95
3.93
51.08
42.0
180.08
160.1
3.33
3.7
12.47
11.11
56.52
42.0
180.08
130.4
3.28
3.8
38.09
13.68
CONCLUSION
By systematically modifying different variables in simulation and in field testing, the
team was able apply Newtonian physics to model rockets in a variety of different scenarios. The
team also discovered how the Tsiolkovsky rocket equation applies to model rocket flights. This
was accomplished by verifying the acceptability of OpenRocket software’s results in predicting
the launch data associated with each flight for each of three models of rockets, as the software
chiefly utilized the rocket equation to make its predictions. The team was able to analyze the
different models of rockets performed in comparison to one another, how increasing mass
decreases the apogee and time of flight, and how thrust data collected from field testing
compared with advertised thrust values from Estes. Ultimately, the team found that, despite their
inability to measure certain variables in the field, such as wind speed, OpenRocket still predicted
the apogee and time of flight of the rockets to some degree of accuracy; the apogee’s average
percent error for the Alpha model rocket was 11.99%, 27.46% for the Viking model, and 14.57%
for the Generic E2X model. The time’s average percent error for the Alpha model rocket was
27.66% , 8.16% for the Viking, and 14.50% for the Generic E2X.
As with many experimental procedures performed in open, uncontrolled systems, there
were several sources of error in performing this lab. Some variables in the field, such as wind
speed, humidity, or air density, could not be measured and did not remain completely constant
throughout the duration of each trial, potentially changing the trajectory of the rocket. This,
along with the fact that the researchers’ positions on the field had no means of measuring a
rocket’s movement along the z-axis, would lead to measured apogees smaller than the distance
actually covered by the rocket. The specifics of each rocket launch were measured using a
stopwatch and two anglers, none of which could measure data to more than three significant
digits, and therefore differences between experimental and anticipated results can be somewhat
attributed to these minor lapses in precision. Additionally, angle and time data was recorded at
the time of parachute deployment, not apogee, as measurements were taken outside and it was
often difficult to see the exact location of the apogee. Different team members were assigned to
different tasks each launch, which may have also contributed to discrepancies in recording data
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in the field. Within OpenRocket, it was sometimes difficult to precisely replicate the models
constructed by the team, as small imperfections in the models, such as paint or glue bubbles or
gaps within the models construction, may have gone unnoticed and were not simulated. In order
to more accurately represent the constructed models, the masses of the rockets in OpenRocket
were overridden to reflect the team’s rocket masses, which may have inadvertently modified the
locations of the centers of mass and altered the flight path of the simulated rockets. Each of
these errors could lead to simulation heights that were higher or lower than the recorded heights,
depending on which variables were not accurately recreated in the software.
Ultimately, these experiments allowed the team to explore the Tsiolkovsky rocket
equation and Newtonian physics, as well as their influence upon rocketry. By utilizing
simulations, field testing, and calculations done by hand, students were able to gain an
appreciable understanding of the fundamentals of model rocketry. This information can be
applied to many aspects of modern rocketry and allows students to participate actively and
intellectually in the ever-expanding field of rocket science.
FUTURE OF ROCKETRY
For the past several years in the United States, leading the charge in space age
applications of rocketry has been the National Aeronautics and Space Administration or NASA.
Created in 1958, NASA has utilized rocketry, among several other disciplines of science and
physics, to launch forward into what is now regarded as “the final frontier”. Space travel has
since been a staple feature of American culture, as NASA continued to pioneer new innovative
technology and research to increase mankind’s general knowledge of the universe as a whole
(16). However, new problems arose within the nation, and general interest in space seemed to
wane, leading to the slow but sure decline in NASA’s presence in research.
After NASA’s funding was cut, the future of rocketry seemed shaky, but recent
developments within space exploration has kept rocketry current and necessitated. However,
NASA is no longer developing new shuttles and rockets, but rather space exploration has been
transferred to the private sector (17); one new private space company is Space Exploration
Technologies, or SpaceX. Its energy is described as “young, hyper-caffeinated”, and reminiscent
of old NASA (17). Its early successes include sending two of its own rockets successfully into
orbit, as well as adding two capsules to the International Space Station (18). The company was
created by Elon Musk, who also happens to be the founder of the electric car company Tesla.
Musk’s ultimate goal is to eventually settle other planets, focusing particularly on Mars (18). He
believes that it is entirely practical to send astronauts to Mars: “It would take six months to get to
Mars if you go there slowly, with optimal energy cost. Then it would take 18 months for the
planets to realign. Then it would take 6 months to get back, though I can see getting the travel
time down to three months pretty quickly” (18). During January 5th’s “Ask Me Anything”
session of Reddit, Musk said that the details of his Mars Colonial Transporter would be revealed
by the end of 2015. This transporter is anticipated to be different than SpaceX’s current Dragon
Capsules and Falcon 9 Rockets, as its goal is to send 100 metric tons of “useful payload” to Mars
(19). Along with the transporter, SpaceX will unveil plans for spacesuits that are not only fully
functional but also aesthetically pleasing. Even though Musk did not specifically say where or
how the spacesuits will be used, it is very possible that they will be used in future Mars
exploration (19).
[1-16]
NASA, however, is still at the forefront of
research, as, for example, New Horizons continues its
mission to capture never-before-seen images of distant
celestial bodies to gather data (Fig. 17). It passed Pluto
on July 17th, 2015 giving the world incredible photos of
Pluto, which are vastly more detailed than any previous
images (20). The spacecraft will continue to study Pluto
until January 2016, when it will then depart beyond the
solar system to the Kuiper Belt. In the Kuiper Belt,
researchers wish to study two bodies, 2014 MU69 and
Figure 17: The New Horizons
2014 PN70, which are about 3 billion miles from Earth. Even
orbiter was an impressive
though NASA has not officially approved to extend the mission to application of rocketry in deep
see both bodies, it is expected that it will continue this exploration space. (20)
in some capacity (20).
While space may still seem relatively unexplored, the field of rocketry presents itself in
stark contrast. Well-known and well-developed, rocket science is in fact so well-understood that
the basic principles of rockets can be explained to young students with little difficulty. Among
the most thoroughly developed of the sciences, rocketry maintains its capacity to play a pivotal
role in the everyday lives of countless people, while expanding its potential for application in
exceptional situations. Although the future remains unclear, it can be predicted that rocketry will
continue to permeate throughout both domestic and global affairs, as its versatility allows the
science to be applicable in various situations. This team project exposed students to the
fundamentals of rocket science, simulations, and model rocketry, thus further expanding the
potential for the expanded use of rockets in the future.
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