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 MurawskiAssistant: 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.

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)

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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 fireworks utilized not one, but two rockets for propulsion (Fig. 2). As the first stage burned out, it would fall back to the earth, making the second stage more efficient and capable of reaching higher altitudes (6).

By 1687, rocketry had truly earned the distinction of a “science”, as this technology was finally established to have capitalized on Newton’s recently formulated third law of motion (3). With the publication of Principia, many were able to visualize the fundamental concepts behind this technology that had been in use for centuries, spurring further understanding and further 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 meters from the launch pad, Goddard’s oxygen and gasoline fueled rocket ushered in a new era of progress in

rocketry (3). He continued to research his liquid fueled rockets, as 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 catalyst for the space race between the United States and the Soviet Union, but it was also the precursor to a space program in the U.S (3).

Figure 2: Schmidlap’s Rocket was the first “step rocket” or a rocket with staging. It used two rockets for propulsion, allowing it to reach very high altitudes. (7)

Figure 3: Goddard’s Rocket utilized liquid fuel, specifically oxygen and gasoline. His research with liquid fuel and parachute systems made him the father of modern rocketry. (9)

Figure 4: Sputnik was the first ever Earth-orbiting satellite. It was launched by the Soviet Union sparking the Space Race. (10)

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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 ispt = 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 topt + dt = mv + mdv - dmev - dmedv + dmev + dmedv - dmeve

(1.3)

Which, after eliminating products of differentials, simplifies to

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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 = dpdt = 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.

Δ p=¿pt + dt - pt = 0mv + 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

dv = -V e

m dm(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∫ 1m

dm

(2.4)After rearranging the logarithmic expression, the negative sign goes away as the initial mass and final mass are flipped in the equation.

∆ v=−ve ln ( mm0

)

∆ v=ve ln ¿)(2.5)

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

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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 pressure, are defined as specific points in a rocket’s design. The center of gravity is the average location of the weight of an object, 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

dvdt

=Tm

−g−12

Cd ρA v2

m (3.2)

This can be simplified by defining two constants C1 and C2 as

Figure 5: A thrust curve demonstrating how the thrust increases rapidly until it runs out of fuel and then thrust decreases.

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C1=Tm

−g

(3.3)

C2=12

Cd ρAm

(3.4)

changing the original equation as dvdt

=C1−C2 v2

(3.5)

After separating variables, this becomesdv

C1−C 2 v2 =dt

(3.6)

This equation can be integrated to obtain

∫ dvC1−C2 v2 =∫dt

1√C1C 2

tanh−1(√ C2

C1v )=∆ t

(3.7)

And, finally, solving for v yields

v=√ C1

C2tanh (√C1C2 ∆ t)

(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=dydt

=√C 1

C 2tanh ¿

∫ dy=√ C1

C2∫ tanh (√C1C2 t)dt

(3.9)

Therefore, the position formula can be expressed through integration as

y=√ C1

C2

ln|cosh (√C1C2t )|√C1C2

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y=ln|cosh(√C1C2t )|

C2(3.10)

This equation can be used to determine the vertical position of a model rocket at any given time during the thrust periodBecause 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.

m dvdt

=−mg−(Cd ρA ) v2

(Cd ρA )=C (a constant )(4)

Through separation of variables and integration, an expression of time as a function of velocity can be created.

−1g ∫ dv

1+ Cmg

v2=∆ t

(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.

y=

m ×ln|cos ( tan−1(√ Cmg vo)−∆ t √Cg

√m )|C

(4.2)

EXPERIMENTAL PROCEDURE

OpenRocketThe 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.

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Experimental LaunchModels 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.

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.

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 6: A comparison between Estes models Alpha (L), Viking (C), and Generic E2X (R) all of which the team constructed and tested.

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.

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DATA & ANALYSIS

The team conducted experiments to provide data in the following three studies:

1. Comparison of apogee in OpenRocket simulation vs. experimental launch2. Acceleration vs Time: Comparison of thrust in Estes vs. experimental launch3. Effect of payload on apogee

Study 1The 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).

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 DataMass of Average Max Simulation Time (s) Simulation % Error % Error

Figure 8: An example graph of a rocket’s flight modeled on OpenRocket. The program showed times of motor burnout, parachute deployment, and apogee.

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Rocket (g) Angle (°) Height (ft) Height (ft) Time (s) Height 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. On average, Viking rockets had a lower mass than the other rocket designs, with their average mass being 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 DataMass of Rocket (g)

Average Angle (°)

Max Height (ft)

Simulation Height (ft)

Time (s) Simulation Time (s)

% ErrorHeight

% 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

*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 DataMass of Rocket (g)

Average Angle (°)

Max Height (ft)

Simulation Height (ft)

Time (s) Simulation Time (s)

% ErrorHeight

% ErrorTime

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

Figure 9: Various numbers and shapes of fins were out on the Viking rockets. Two rockets had 5 fins, one had 4, and the last had 3.

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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).

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 2This 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.

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.

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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:

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)=905.613t3. We could then use this equation to determine the thrust vs. time equation for each launch as the

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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/s3.

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/s3, which is equivalent to approximately 276.031 m/s3. 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-D. 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=b2mt2. Since v=dydt, so we can write the rearranged equation dy=b2mt2dt and integrate both sides to obtain the equation y=b6mt3. 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/s3, 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%.

Figure 12: A8 Thrust graph given by Estes that was used to calculate the acceleration of datapoints. (15)

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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 3This 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 average percent error being 13.72% 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).

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 DataMass of Rocket (g)

Average Angles (°)

Max Height (ft)

Simulation Height (ft)

Time (s) Simulation Time (s)

% ErrorHeight

% 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

Figure 13: Line graph that shows relationship between mass and apogee in the experimental launches and the simulation.

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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 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.

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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).

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 2014 PN70, which are about 3 billion miles from Earth. Even though NASA has not officially approved to extend the mission to see both bodies, it is expected that it will continue this exploration 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.

Figure 17: The New Horizons orbiter was an impressive application of rocketry in deep space. (20)

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