Rocket Lab
Albi Belliston
Academy for Math, Engineering, and Science
Period: 4B
Mr. Hendricks
Abstract:
In the lab conducted, rockets were launched and had the heights predicted and compared
to the actual values. Through various calculations done, heights were predicted as to how high
the rockets would go. Engine Thrust Analysis, Drag Force, and Numerical Model calculations
were done to find the heights. Rocket engines were placed on a cart with a Digital Force Gauge
to find the Average Force exerted and Total Impulse to predict the type of engine. Engine-less
rockets were then placed inside a Wind Tunnel and data was used to find the Drag Force
Coefficient Constant. Numerical Iteration was then used to find the predicted heights on an
Excel Spreadsheet using a time interval of .1 seconds, making the results viable, considering
that the time interval was small enough; otherwise Differential Equations must be used.
Rockets were then launched and 3 angles were taken to find the actual heights. The predicted
heights for the engines used are 324.46 meters for the white rocket with a C6 engine, 160.99
meters for a white rocket with a B6 engine, 237.84 meters for a blue rocket with a C6 engine,
and 89.73 meters for a blue rocket with a B6 engine. Only the white rocket was launched and
the actual heights are 50 meters for the B6 engine and 130 meters C6 engine. Results had to be
rounded due to estimation on most other calculations done.
Introduction:
The purpose of the project was to meaningfully launch rockets by applying knowledge to
real-world situations in the form of Rocket Science and Physics. A few terms that will be
needed to understand the project are Kinematics, Dynamics, Impulse, Momentum, Drag Force,
Drag Coefficient, and Rocket Engine terminology.
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Kinematics – the study of motion.
Dynamics – the study of how forces affect motion.
Impulse – force as a function of time, or FΔt.
Momentum – the product of mass and velocity of an object, or P=mv.
Drag Force – the resistance to motion through a medium (i.e. air).
Drag Coefficient – a scalar quantity used to determine the Drag Force on an object.
Rocket Engine terminology – _ _ - _ (i.e. B6-4). The letter determines the Total Impulse
of that engine. The first number determines the Average Force the engine exerts. The
final number determines the amount of time (in seconds) before the parachute is released
(not needed).
Numerical Iteration is the process used to determine the type of engine. It is the study of
algorithms that use numerical approximation. Because there are many things that affect how
high the rocket goes, Numerical Iteration must be used or more complex math is needed.
Numerical Iteration also only works as long as the change in time that is used is small enough.
The derivation of the impulse momentum theorem (FΔt = ΔP) is as follows. The
definition of Impulse is I= FΔt. The definition of Force is F=ma. Insert that in for “F” in FΔt.
Now, I=maΔt. The definition for acceleration is a= Δv/Δt. Insert that in for “a” in I=maΔt.
Now, I=(m)(Δv/Δt)( Δt). The “Δt’s” will cancel out, therefore I=mΔv. Δv is defined as vf-vi.
Therefore, I=m vf-mvi . Momentum is defined as P=mv, which changes the equation to I=Pf-Pi.
So, I=ΔP, or FΔt = ΔP. The derivation is complete.
Engine Thrust Analysis:
The materials needed to conduct this section are a T.I. 83/84 Calculator with a CBL to
record the results, a Digital Force Gauge to measure the force exerted, a Datamate calculator
program to run the experiment, a model rocket engine to light, an ignition mechanism with plug
to attach to the rocket to ensure “launch,” appropriate alligator clips that will be attached to a
battery, a battery with sufficient strength to launch the engine, and a box attached to cart with
negligible friction wheels to hold the engine in place while it exerts the force being measured,
without falling off the track.
Procedure:
Object needed were obtained and set up as to run the experiment properly. Time Graph
Settings on the calculator were adjusted. Time intervals were set to every .1 seconds.
Therefore, 50 samples (to cover 5 seconds total) would be used and calculated.
The following steps were performed in order to set up the experiment. The battery was
hooked up to the engine and the engine was inserted securely into the box on the cart and the
Digital Force Gauge was attached. The Gauge was assembled to the calculator and triggering
was activated. Settings included a Decreasing Trigger, Threshold: -1. These settings were
placed to ensure that the trigger is “higher” (although the values will show up as negative, just
take the absolute values) than the “signal noise.” The Datamate program was run and the
battery was activated. When the engine finished going off, the tables with the thrust data
showed up in L1 and L2 on the calculator.
The values were recorded and the Total Impulse and Average Force was found to
determine the kind of engine.
Air resistance should be accounted for, if possible. If not, then this fact must be included
in the results.
Results:
Time (s) Force (N) Impulse (Ns)
0.6
2.35
0.235
0.7
8.39
0.839
0.8
5.64
0.564
0.9
4.4
0.44
1
4.01
0.401
1.1
3.86
0.386
1.2
3.84
0.384
1.3
3.8
0.38
1.4
3.77
0.377
1.5
3.74
0.374
1.6
3.79
0.379
1.7
3.81
0.381
1.8
3.8
0.38
1.9
3.78
0.378
2
3.77
0.377
2.1
3.84
0.384
2.2
3.75
0.375
2.3
3.86
0.386
2.4
3.89
0.389
2.5
0.496
0.0496
2.6
0.031
0.0031
Conclusion:
For the Total Impulse, 7.8596 was found, which is closest to a 10, not 5. For the Average
Force, 3.9293 was found, which is close to 4. Concluding, the engine was a C4 Engine. Air
Resistance was not accounted for, but will be included in later segments of the project. The
Total Impulse found was far enough away from both the B and C engine values that it is merely
a guess. Since the value calculated for the Total Impulse was ~.35Ns away from the exact
middle between the Impulse of the B and C engines, it is very hard to conclude that it is an
accurate engine.
Drag Force:
The materials needed for this section are as follows. A wind tunnel was needed in order
to generate some measurable drag force and solve for the Drag Coefficient. An anemometer/
wind velocity gauge was needed to find the wind velocity to plug into the drag force equation.
A protractor was also needed to measure the angle at which the rocket was flying. Lastly, a
string to hold the rocket in place was needed to ensure that the rocket did not fly all over the
wind tunnel, or else the Drag Coefficient would have been incorrect, ruining all other results in
the project.
Procedure:
Using a scale, the mass of the rocket was found in Standard Physics units (kg). Next, the
Rocket was placed inside the Wind Tunnel and secured with a string. The Wind Tunnel was
then turned on and using the protractor, the angle at which the Rocket is hanging/ flying was
found.
Now, the Drag Force (Fd) must be found by using the equation Fd=mgtan θ, the Drag
Force equals the mass of the rocket times the Force of Gravity (9.8) time the Tangent of the
Rocket’s angle in the Tunnel. To derive the equation, see the Derivation Segment of this report.
Finally, the Wind Velocity Gauge was inserted into the Wind Tunnel and it was turned on
again. The Wind Velocity was then recorded in Standard Physics units (m/s). The other
derived equation (Kd=Fd/v2), the Drag Coefficient Constant equals the Drag Force divided by
the Wind Velocity squared, was used to find the Drag Coefficient. Results were then recorded.
The honeycomb structure inside the wind tunnel was to ensure that Laminar Flow was the
kind of flow generated inside the tunnel. Without the honeycomb structure, turbulence would
be the type of flow that we would get, which would result in an incorrect Drag Force and Wind
Velocity.
Results:
The wind velocity found was 36.3 meters per second. The angle of the rocket was 25 o.
The mass of the rocket was .061kilograms. The drag force calculated was .2788 Newtons. The
drag coefficient was then found to be .0002.
Calculations:
Fd=mgtan θ
Fd=.061x9.8xtan(25)
Fd=.2788N
Kd=Fd/v2
Kd=.2788/36.32
Kd=.0002
Derivations:
Numerical Model:
It is very difficult to find the exact values of predicted heights. There are two options to
find said heights. Either Differential Equations or Numerical Iteration can be used. Numerical
Iteration will give an approximate answer, but the number given is so close to the real answer
that it really doesn’t matter. It is okay to use this method of Numerical Iteration as long as the
change in time (Δt) that is used to calculate the heights is small enough, usually .1 seconds or
less.
Procedure:
Using the provided Microsoft Excel spreadsheet, the height predictions for 4
circumstances will need to be found: the White Rocket with a B6 Engine, the White Rocket
with a C6 Engine, the Blue Rocket with a B6 Engine, and the Blue Rocket with a C6 Engine.
7 Variables will need to be known in order to find the predicted heights: the masses of the
Blue and White Rockets, the masses of the B6 and C6 Engines, the Drag Force Coefficient
Constant (Kd), and the Thrust data for the B6 and C6 Engines.
3 things will now be needed to be entered into the spreadsheet: the Drag Coefficient (Kd)
of .0002, the Thrust Data for the engine put in the Rocket, and the combined mass of the
Rocket, found by adding the mass of the Rocket and the chosen Engine, all of which was
entered into the appropriate places on the spreadsheet.
After plugging in the values for all 4 situations, the highest height for each situation was
found, usually on the same row as the last positive or first negative Average Velocity found on
the spreadsheet.
The formulas used on the spreadsheet can be found under each row category.
To simulate what it would be like without air resistance, 0 was plugged in for the drag
coefficient on the spreadsheet to simulate that there would be no resistance. Doing so resulted
in the maximum height not being found on the spreadsheet, meaning that the rocket would
continue to fly higher and higher.
Predicted Heights:
White Rocket with C6 Engine: 324.46 meters
White Rocket with B6 Engine: 160.99 meters
Blue Rocket with C6 Engine: 237.84 meters
Blue Rocket with B6 Engine: 89.73 meters
Flight Results:
The actual heights were found by placing 3 people with protractors exactly 50 meters
away from the launch point, roughly 120o apart from each other. The angles at which the
rockets were at during each launch were found using the protractors. However, only the white
rocket was launched. Several people must be used because the rocket did not fly in a straight
line, so the average of each angle found was taken to get the most accurate results.
Drawing of Triangle:
The trigonometric function used was tangent, since the distance of 50 meters was given,
and the average angle was used, the hypotenuse of the above triangle was not needed or
relevant, because the height at which the rocket was flying was what was being calculated.
Also, the heights of the people had to be added onto the total result because the angles were
measured ~1.5 meters above ground level.
The angles for the heights were 50o, 30o, and 49o, with an average of 43o for the B6
engine, and 55o and 59o for the C6 engine, with an average of 57o. The third angle was not used
because the rocket flew above the other person’s head.
Actual Heights:
B6: 48.1 meters, rounded to 50 meters.
C6: 134.85 meters, rounded to 130 meters.
Conclusion:
The predicted height for the B6 engine was 160.99 meters, and the actual was 50 meters,
with a difference of ~111 meters. The predicted height for the C6 engine was 324.46 meters,
but the actual was only 130 meters, making a difference of ~195 meters. Since the blue rocket
was never recovered, it could not be launched.
The heights did not match up perfectly. This could be due to several factors. Because
the cart with the engine in it fell off the track, the force measured was inaccurate. This factor
threw off the conclusion that it was a C4 engine, when in fact it was indeed a C6 engine. The
drag coefficient might have been thrown off by the simple fact that the wind tunnel might not
have produced the same wind speed when it was turned on to measure the angle of the rocket
and when its wind velocity was measure by the anemometer. Overall, many things were
rounded due to uncertainty and only one significant figure was used. The angles at which the
rocket was at in the end were not very accurate due to simple human error. Finally, more
accurate equipment could have solved most of the problems.
If the project were to be done again, more accurate equipment could be used to allow for
more accurate results by using more significant figures and less rounding. I force gauge could
have been attached to the wind tunnel to give the exact Drag Force without more math being
used and throwing off the results further. If all of these were to be done, the predicted heights
would most likely be significantly closer to that of the actual heights.
Reflection:
I gained a much better understanding of physics by doing this project because I got to do
nearly every calculation by hand, allowing me to go through step by step to solve the problems I
encountered. The project also allowed me to be more independent because I was doing problem
solving and ultimately came to a pretty good conclusion in which I was fairly confident in my
results.
Some of the difficulties I encountered while doing this project was trying to figure out
what the type of engine used in the first section was. I should have used area under the curve to
find the average force, instead of just simply adding them all together and taking the average,
however, the cart with the engine did fall off the track, thus lowering the readings for the
average force and giving me a C4 engine result, instead of a C6 engine. To overcome that, I
simply accounted for that in my conclusion.
Lastly, this project was an excellent example of critical thinking and problem solving that
helped me accomplish a goal, find the predicted heights and compare to reality.