Mechanics of a projectile fired through a
vacuum cannon
Martin Macák & Lothar Nygren
Supervisor: Ulf Christiansson
Stockholm, Viktor Rydbergs Gymnasium
Diploma project
2019-02-21
1
Abstract
In this report the phenomena and build of a vacuum cannon was presented together with a
theoretical model of the velocity when a projectile is fired out of it. The model states cross
sectional area of the cannon as a factor of velocity, but does not explore the cross sectional
area of the projectile, which was the aim of this report. Using a high speed camera, the
velocities of projectiles with different areas could be analysed. The results indicated a
correlation between larger projectile cross sectional area in relation to pipe cross sectional
area and velocity. However, the exact relationship could not be stated due to limited sample
sizes and experimental inaccuracies. This result fits the hypothesis that there is more blow-by
wind acting in retarding motion for projectiles with smaller cross sectional area. Blow-by
wind is also hypothesised to be the cause of maximum velocity near the middle of the cannon
opposed to the cannon muzzle that the theoretical model predicts.
2
Table of contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
Introduction ………………………………………….…………………………… p. 2
Theory ………………………………………….………………………………… p. 3
Method …………………………………………………………………………… p. 4
3.1.
Materials …………………………………….…………………………… p. 4
3.2.
Procedure…………………………………….…………………………… p. 4
3.3.
Safety remarks …………………………………………………………… p. 6
Results ………………………………………….………………………………... p. 7
4.1.
Maximum velocities …………………………………….……………….. p. 7
4.2.
Characteristic shot …………………………………….……………….… p. 9
Discussion ………………………………………….……………………………. p. 10
5.1.
Analysis…………………………………………………………………... p.10
5.2.
Experimental observations ………………………………………………. p. 10
5.3.
Experimental validation …………………………………………………. p. 11
Conclusion ………………………………………….…………………………… p. 12
Acknowledgment ………………………………………….…………………….. p. 12
List of sources ………………………………………….………………………... p. 12
Appendix ………………………………………….……………………………... p. 13
1.
Introduction
A vacuum cannon is a ballistic apparatus that uses atmospheric pressure in order to
accelerate a projectile. The cannon consists of a long pipe, with sealed ends, which is
connected to a vacuum pump. When almost all air inside the pipe has been removed and one
of the seals is broken, air at atmospheric pressure will rush in to move a projectile ahead of it.
With a pipe long enough, and a vacuum quality good enough theoretically the projectile can
be accelerated near 300 m/s. However the “first-order model” brought forward by Ayars and
Buchholtz (2004) does not take all possible variables into account, and the actual velocity for
a projectile turns out to be less.
In ​The mechanics of a ping-pong ball gun French, Gorrepati, Alcorta, and Jackson (2008)
examine, among other factors that affect the projectile’s velocity, the blow-by wind in front
of the projectile. With FLUENT, a computational fluid dynamics software, they conclude that
blow-by wind is evident with a supersonic airflow. However the flow’s effect on the
projectile’s velocity was not evaluated.
The aim of this report is to analyse projectiles’ cross-sectional area, relative to the
cross-sectional area of the pipe, and how area differences affect velocities of a projectile
when fired within a vacuum cannon.
3
2.
Theory
In ​Analysis of a vacuum cannon (​ Ayers & Buchholtz 2004) the muzzle velocity of a
projectile fired from a vacuum cannon is analysed. At the most basic level the force on the
projectile can be assumed to be dependent on the atmospheric pressure subject to the
projectile, the projectile’s cross-sectional area and mass with the premises that the projectile’s
acceleration is constant.
However the report states that the force resulting from the pressure difference between the
outside and inside of the pipe does not only accelerates the projectile, but also a growing
column of air behind the projectile, the “first-order model”. A formula as function of pipe
length can be deduced if it is assumed that the projectile seals the pipe perfectly, and effects
due to friction and air compressibility are negligible. The velocity (​v​) is shown to be
dependent on the length of pipe (​x)​ , pressure difference between air in and outside the pipe
(ΔP), the air density (ρ) and a constant (λ). (​See Appendix A for complete deduction of
formula).
v (x) =
√
ΔP
ρ
*
x
x+λ
*
√1 +
2λ
x
(1)
The constant lambda is in its turn dependent on the mass of the projectile (m), cross-sectional
area of the pipe (A), and the air density (ρ). The constant can effectively be replaced with
m
λ = ρA
for a complete formula.
v (x) =
√
ΔP
ρ
*
x
m
x+ ρA
*
√1 +
2m
xρA
(2)
Nevertheless the assumption that the projectile seals perfectly is only an assumption and air
will move ahead of the projectile, also known as blow-by wind. This will create a high
pressure area in front of the projectile causing aerodynamic drag that retards the projectile
(French et al. 2008). With different cross-sectional areas of fired projectiles’ the hypothesis is
that aerodynamic drag will be differ due to different volumes and turbulence of the blow-by
wind, resulting larger projectiles with less blow-by wind moving faster.
3.
Methods
3.1.
Materials
4
-
Transparent acrylic pipe (2000mm)
Hose nozzle
Ultimaker 2+
Add:north E-PLA 3D-printing filament
Cura 3D printing software
Vacuum pump
Wireless electronic pressure sensor
High-speed camera
Logger Pro analysing software
Scale
Packing tape
Duck tape
Burst stick (sharp metal stick)
Saw
Sand paper
Rasp
Drill
2-part Epoxy glue
3.2.
Procedure
An 909 mm long pipe was obtained, and the edges of the pipe were refined with a rasp and
sandpaper, leaving polished edges. A hole was drilled 78 mm away from one end of the pipe,
and hose nozzle was inserted 2.5 mm deep and fitted with two-part Epoxy glue to obtain a
tight fit. A vacuum pump hose was connected to the hose nozzle on the pipe with hose
clamps. The aim is to create an airtight space free from leakage so that a vacuum can be
created.
Four spherical projectiles (See Appendix B) with different diameters and thus different
cross-sectional areas (Table 1) were 3D-printed using an Ultimaker 2+ . This is done due to
cross-sectional being a key variable analysed in this report. To keep mass as a constant, the
densities of the projectiles were varied with the infill range on the 3D-printer. To measure
discrepancies in mass, the projectiles were weighed on a scale.
Projectile Color
Radius
(mm)
cross-sectional area (m​2​)
Percentage of pipe Mass (g)
cross-sectional area
1
32.00
3.217 * 10​-3
75.82
Green
53.65
5
2
Red
35.80
4.026* 10​-3
94.90
51.25
3
Black
36.35
4.151 * 10​-3
97.83
52.50
4
White
36.50
4.185 * 10​-3
99.18
51.93
Mean:
52.33 ±1.32
Table 1:​ Technical specifications of projectiles
In order to create a vacuum, low pressure inside the pipe, the ends of the pipe were each
sealed with two layers of packing tape working as end and entrance cap, and the attached
vacuum pump was switched on. The pressure inside the sealed pipe was measured with with
a wireless electronic pressure sensor. Measurements indicated (See Appendix C) that with
certainty the lowest pressure had been obtained by 60 seconds from turning on the vacuum
pump. The pressure obtained in the vacuum cannon was 2230 ±296 Pa, resulting in a pressure
difference of 97689 Pa with equal error margin.
For measuring velocity of projectiles the cannon was placed on the ground, and a high speed
camera was set up perpendicular to the pipe framing the starting position of the projectile to
the end cap of the pipe. A ruler parallel to the pipe was placed in the frame for scale in video
analysis. In order to shoot, a projectile was placed in its starting position, 163 mm from the
entrance cap, and a vacuum was created by the process described above. After 60 seconds the
entrance cap was burst using a burst stick, allowing the projectile to rush out at the opposite
side of the pipe. The shot was recorded at 1200 fps with the high speed camera, and the
process of loading and shooting was repeated three times for each projectile. Projectile
velocity at different times were analysed with Logger Pro analysis software. In Logger Pro
the speed is analysed by manually pointing at the projectile’s center at each frame, and speed
discrepancy emerges thus different manual marks, but through the course of analyse the
discrepancy did not exceed 3%.
6
Illustration 1:​ Vacuum cannon and its supplements arranged to shoot a projectile (See appendix D)
3.3.
Safety remarks
The maximum theoretical velocity of a projectile is ~113 m/s, and this gives a large kinetic
energy. Cautionary measures need to be taken to minimize the dangers associated with such
large energies. A barrier of layered cloth and a plastic box was placed behind both caps to
stop projectiles. Note that an accidental burst of the end cap can project the projectile out of
the entrance cap. In case of accidental bursts it is also not recommended to stand in front of
either end of the vacuum cannon while the vacuum pump is on. For this reason a “burst
stick”, a sharp edge connected to a long stick, is used to burst the entrance cap safely. As the
caps are burst, a loud sound occurs, and protective earmuffs were worn. Safety glasses were
worn due to possibility of breakages and fragments with large velocities and large forces in
action.
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4.
Results
4.1.
Maximum velocities
Maximum velocity for every shot obtained after analyse with Logger Pro was recorded in
Table 2, and further plotted with error bars as a scatter diagram in Figure 2. In the graph each
mark’s color corresponds to the fired projectile’s respective color (Note that white projectile
is painted grey), and the maximum velocities are juxtapositioned to a orange line
demonstrating theoretical maximum velocity for the used cannon’s specific length.
Additionally the velocity formula is plotted in Figure 1 with a mark at 112,6 m/s
corresponding to the cannon’s specific length. Figure 3 shows the measured velocities’
percentage of theoretical maximum velocity dependant on the specific projectile
cross-sectional areas’ percentage of the cannon’s cross-sectional area in order to demonstrate
deviation from velocity formula.
Maximum velocity (m/s)
Projectile Color
Percentage of cross- Series 1
sectional area
Series 2
1
Green
75.82
27.7
21.3
2
Red
94.90
56.1
39.6
3
Black
97.83
65.8
48.5
4
White
99.18
57.8
48.9
Table 2:​ ​Maximum velocity for the projectiles
Figure 1:​ Muzzle velocity as function of cannon length
8
Figure 2:​ Projectiles’ maximum velocities dependant on their cross-sectional area
Figure 3: Maximum velocities of projectiles relative to theoretical maximum velocity as function of projectile
cross-sectional area relative to pipe cross-sectional area
9
4.2.
Characteristic shot
With a high speed camera and a clear pipe it was possible with Logger Pro to analyse
momental velocities of the projectile inside the pipe. Picture 1 shows the trajectory between 5
to 15 ms of the red projectile fired in data first measurement series (Table 2). Note that top
velocity occurs in second frame after 9 ms, and end cap has burst in third frame before
projectile reached it at 15 ms. Figure 3 s​ hows the projectile’s velocity as function of time
after analysis in Logger Pro, and the graph’s appearance is characteristic for all shots (see
Appendix E). Selected data points in the graph have been paired with the analysed frames
below to highlight relevant observations.
Picture 1​: Red projectile’s position after 5, 9 respectively 15 ms shot in series one
Figure 4:​ Velocity as function of time for the red projectile’s shot in series one
10
5.
Discussion
5.1 Analysis
The results show that the largest velocity was obtained by the second largest projectile,
however in series 2 the projectiles’ larger maximum velocities correspond to their larger
cross-sectional area. The three largest projectiles that all exceed 94% of cross-sectional area
aslo travel significantly faster than the green projectile that only took up roughly 75% of the
total cross-sectional area. There seems to be correlation between cross sectional area of
projectile and velocity, as well as a correlation between deviation from theoretical velocity
depending on ratio of cross sectional area compared to the pipe. The results can be explained
by a greater presence of blowby wind that creates turbulence which in turn slows the smaller
projectile down, also known as aerodynamic drag. The correlation in figure 3 cannot be
deemed either linear or exponential due to few data points, but its presence is strengthened by
all the higher and lower data points respectively belong to the same measurement series.
By looking at a characteristic shot (Figure 4) and comparing it to the velocity formula as a
function of distance travelled inside a vacuum pipe (Figure 1) it can be revealed that there is
conflict between the two. The results show that the maximum velocity occurs approximately
in the middle of the cannon, and not at the muzzle as the model predicts. It can further be
seen that the projectile is significantly retarded near the end cap and thus further lowers the
muzzle velocity. This could be due to blow-by wind creating an air pillar forming in front of
the projectile which creates aerodynamic acting as a cushion. None of the projectiles tested
were a perfect seal to the pipe which accounted for some blow-by wind in all measurements.
As the premisses of the theoretical formula are not entirely satisfied, it is unclear if this effect
is previously unknown or if it occurs due to experimental errors.
5.2 Experimental observations
During the time the shots was performed a few observations could be noted. The cannon
recoils when fired thus was it taped to the ground to minimize the recoil effect. When the
entrance cap was pinched it randomly burst in different manners (​see appendix F).​ If only a
small part of the entrance cap bursts it leads to air coming in at an angle which in turn could
lead to a considered bad shot where to projectile bounces up and down during its trajectory,
or does not move from its starting position. The cap can also leak or burst before a full
vacuum is created, thereby annulling the shot.
Furthermore the projectile in some instance does not fly through the end cap, but instead
bounces back from the tape. In the instances where it flies through the end cap, the muzzle
velocity is not greater than 23m/s. After calculation with Euler’s method, a spherical
11
projectile with the mass and cross-sectional areas such as the ones used will be retarded to at
least 19 m/s at 4m distance from the muzzle, thereby passing as legal in Sweden.
5.3 Experimental validation
There is considerable scatter in the data measurements between the two series. This could be
due to factors such as friction between projectile and pipe, as well as in the friction in the
form off drag between the projectile and the remaining air in the pipe. There is also the fact
that all measurement in the second series gave lower velocities than the first one. As all of the
measurements of the second series were subsequent to the first measurements, it is possible
that the vacuum quality sunk due to leakage between the series.
The premisses for the theoretical maximum velocity stated that a projectile seals the pipe
perfectly. None of the projectiles did so, and the theoretical maximum velocity should be
seen as one for a ‘perfect’ projectile.
Several experimental factors may have affected the velocity. The starting position of the
projectile was located 8 cm from the entrance cap due to the hose nozzle sticking in. The
formula for theoretical velocity assumes that the projectile starting position is at the entrance
cap. This means that the projectile did not have the whole length of the cannon to accelerate.
Additionally the initial force that hits the projectile could be lower than the initial force at the
entrance cap as some of the force been used to accelerate the 8cm long air column in front of
the projectile.
The bursts of the entrance cap were random and leave a portion of the tape around the edges
that may interfere with the incoming air flow. As the projectiles were of different size and
positioned in the bottom leaving their centers positioned at different distances from the pipe’s
center thus incoming air striking projectiles differently. Additionally the hose nozzle sticks
2.5 mm into the tube which could similarly interfere with the incoming air, and both
interferences could lower the velocity of the projectile.
The experiment was conducted with a small sample size of projectiles and only two
measurement series. As stated above, there are many factors that could affect the velocity of
the projectiles, and it is thus difficult to draw concise conclusion. This is especially true of of
the three largest projectiles as their velocity ranges were close. A correlation between the
examined factors seems to exist, but because of the small sample size it cannot be pinpointed
with certainty.
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6.
Conclusion
The results indicated that the projectiles did not reach their theoretical velocity maxima. The
results further indicated that the smaller the cross sectional area in relation to the one of the
pipe, the farther the velocity deviates from its theoretical value. This seems to play a larger
significance the closer the cross sectional area of the projectile is to the total cross sectional
area of the pipe. Due to small sample sizes and some experimental inaccuracies the
relationship between the cross sectional area of projectile in comparison of the pipe cannot be
quantified as linear or exponential, and this could be an area of further investigation. An
additional finding was that the maximum velocity of the projectile, contrary to previous
studies, occurred inside the pipe’s middle and not muzzle.
7.
Acknowledgements
The authors would like to acknowledge the House of Science for the use of their equipment
and lab space, and special thanks to Henrik Åkerstedt for his unconditional guidance and
doctor Stefan Åminneborg for his eagerness and support throughout the work at the House of
Science.
8.
List of sources
Ayers, E. Buchholtz, L. 2004. Analysis of the Vacuum Cannon. American Journal of Physics
72 (7): 961-963.
French, R.M. Gorrepati, V. Alcorta, E. Jackson, M. 2008. The mechanics of a ping-pong ball
gun. Experimental Techniques 32 (1): 24-30.
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9.
Appendix
A. Deduction of velocity formula
The force (​F​) resulting from the air pressure, equivalent to the pressure difference (​ΔP​)
multiplied by the cross-sectional area of the pipe (​A)​ , accelerates both the pillar of air and the
projectile. Both entities will have momenta (​P)​ , equal to their common velocity multiplied
with their respective mass. For the projectile mass (​m​) it is constant, while the air pillars mass
increases together with its volume. Specifically the mass of the air pillar is its density (​ρ​) and
volume, the cross-sectional area of pipe multiplied with the distance that the projectile has
travelled (​x​).
F = ΔP A
(5)
P = (mprojectile + mair )v = (mprojectile + ρxA)v
(6)
The acting force is equivalent to the derivative of the combined momentum. The resulting
equation can then be integrated with respect to time (​t​). For convenience the cluster of
m
constants ρA
can be replaced by ​λ i​ n equation (10).
F =
dP
dt
(7)
d((m+ρxA)v)
dt
t
t
d((m+ρxA)v)
ΔP Adt =
dt
t0
t0
ΔP A =
∫
∫
(8)
× dt
(9)
ΔP At = (m + ρxA) × v
= (m(1 + ρxA
)) × v
m
= (m(1 + λx )) × v
(10)
Integrate with respect to time, multiplying with ​dt.​ Note that velocity is the derivative of the
distance travelled.
t
t
t0
t0
∫ (ΔP At) dt = ∫ {m(1 + λx )v} dt
1
2
× ΔP At2 =
t
∫ {m(1 + λx )} v × dt
t0
t
= ∫ {m(1 +
t0
x
)
λ
} dxdt × dt
(11)
14
x
= ∫ {m(1 +
x0
1
ΔP At2
2
x
)
λ
} dx
(12)
x2
)
2λ
= m(x +
(13)
For later convince, t​2​ is expressed using​ a0 =
t2 =
=
=
2m
x2
(x + 2λ
)
ΔP A
2mx
(1 + 2λx )
ΔP A
2x
(1 + 2λx )
a0
F
m
=
ΔP A
m
.
(14)
The equation contains the variable x in quadratic form, which can be solved for x, eliminating
the negative root.
2
x2 + 2λx − λa0 t = 0
(15)
√ λ + λa t
= λ (√1 +
− 1)
2
x(t) = − λ +
0
2
a0 t2
λ
(16)
The function is derivated with the chain rule, as velocity is the derivative of position
v (t) =
=
dx
dt
a0 t
√
1+
=
=
√
a0 t2
λ
a0
a0
λ *
√
λ +1
a0 t 2
√a0 λ
√
1+
(17)
λ
a0 t2
The nominator can be replaced by
the right side of equation (14).
v (t) =
=
=
√ΔP
√
√ρ*
1+
√ΔP
√
√ρ*
λ
a0 t2
λ
x)
0* a2x (1 + 2λ
0
1+ a
√ΔP
√1+
√ρ*
λ2
x(2λ+x)
√
ΔP
ρ
.
based on the the definition of ​a0​ .​ t2 can be replaced
15
√
=
√
=
√
=
√
=
ΔP
ρ
*
ΔP
ρ
*
ΔP
ρ
*
ΔP
ρ
*
√
x(2λ+x)
2λx+x2 +λ2
√
√
x2 ( 2λx + xx )
x
x+λ
*
(x+λ)2
x2 ( 2λx + xx )
(x+λ)2
√1 +
2λ
x
(18)
B. 3D-printed projectiles arranged smallest to largest
C. Pressure data
During the test the vacuum pump was turned on at t=0, turned off at t=85, and the entrance
cap was burst at t=160. The graph below indicated that at t=60 minimal pressure of roughly
2230 Pa was reached.
D. Vacuum cannon and its supplements arranged to shoot a projectile
16
E. Characteristic shots
17
The graphs color is coded by projectile color with the exception of grey graph corresponding to white projectile
18
F. Random burst of entrance caps