![Calculated penguin's rectal pressures Pt with and without the coefficient of the contraction C, as functions of the viscosity η. The dash-dotted curve represents the result of Ref. [2]. For comparison, we also show P0.](https://www.researchgate.net/profile/Hiroyuki-Tajima-3/publication/342655211/figure/fig3/AS:909052053823488@1593746307398/Calculated-penguins-rectal-pressures-Pt-with-and-without-the-coefficient-of-the_Q320.jpg)
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arXiv:2007.00926v1 [physics.bio-ph] 2 Jul 2020
Projectile Trajectory of Penguin’s Faeces and Rectal Pressure Revisited
Hiroyuki Tajima1and Fumiya Fujisawa2
1Department of Mathematics and Physics, Kochi University, Kochi 780-8520, Japan
2Katsurahama Aquarium, Kochi 781-0262, Japan
(Dated: July 3, 2020)
We discuss a trajectory of penguins’ faeces after the powerful shooting due to their strong rectal
pressure. Practically, it is important to see how far faeceses reach when penguins expel them from
higher places. Such information is useful for keepers to avoid the direct hitting of faeceses. We esti-
mate the upper bound for the maximum flight distance by solving the Newton’s equation of motion.
Our results indicate that the safety zone should be 1.34 meters away from a penguin trying to poop
in typical environments. In the presence of the viscous resistance, the grounding time and the flying
distance of faeces can be expressed in terms of Lambert Wfunction. Furthermore, we address the
penguin’s rectal pressure within the hydrodynamical approximation combining Bernoulli’s theorem
and Hagen-Poiseuille equation for viscosity corrections. We found that the calculated rectal pressure
is larger than the estimation in the previous work.
PACS numbers: 01.40.-d,47.85.Dh,87.15.La
I. INTRODUCTION
Penguins, which are aquatic birds living mostly in the
Southern Hemisphere [1], strongly shoot their faeceses
towards their rear side [2]. It is believed that this is be-
cause penguins avoid getting the faeces on themselves as
well as the nest. Although such a tendency is not lim-
ited to penguins and can be found in the case of other
birds, these bombings sometimes embarrass keepers un-
der breeding environments like an aquarium. It is practi-
cally important to know how far their faeceses reach from
the origin. Such information would save keepers from the
crisis. It would also be helpful for a newcomer guidance
for keepers to avoid such an incident.
The flying distance of penguin’s faeces reaches about
0.4 m even on the ground. Since a typical height of a
Humboldt penguin is given by 0.4 m, this distance cor-
responds to the situation that if a human being whose
height is 1.7 m tries to evacuate his/her bowels, the ob-
ject could fly to 1.7 m away. Therefore, one can imme-
diately understand that penguin’s rectal pressure is rel-
atively much strong compared to that of a human kind.
In the pioneering work of Ref. [2], it is reported that this
actual pressure could range from 10 kPa to 60 kPa for
relevant values of the faeces viscosity and the radius of
the bottom hole.
However, in Ref. [2], the projectile tra jectory of the
faeces is not taken into account but the horizontal dis-
tance is employed to estimate the fluid volume. In gen-
eral, the actual trajectory would be longer than the hor-
izontal distance during the pro jectile motion as shown
in Fig. 1. In addition, since the ejection angle is not al-
ways horizontal and penguins sometimes shoot them out
from higher place under the breeding environment, it is
important to consider such a motion of faeces in a more
general manner. The rectal pressure is regarded as an
impulse force to accelerate the faeces up to the initial
velocity in Ref. [2]. For non-viscous fluids, we can use
Bernoulli’s theorem which is related to the energy con-
h
d
v0 = (v0x,v0y)x
y
O
FIG. 1: Configuration for a penguin trying to defecate to-
wards his/her rear side. A penguin stands on the rock with
the height hfrom the ground. We parameterize the ejection
angle θwith the initial velocity v= (v0x, v0y) We estimate a
flying distance dof the faeces from the origin O.
servation. To estimate the viscosity effects on the rectal
pressure, Ref. [2] assumed the Hagen-Poiseille flow in the
air, which is laminar flow of Newtonian liquid in a cylin-
drical pipe geometry. A similar approach for uniaxial
urinary flow was employed to estimate the duration of
urination of various animals in Refs. [3, 4].
In this work, we calculate the maximum flying distance
of penguin’s faeces from a high place, which is relevant
for several breeding environments. Such a projectile tra-
jectory is described by Newton’s equation of motion. We
assume that the upper bound for the flying distance can
be obtained by the equation of motion in the absence of
the air resistance. Moreover, we revisit the rectal pres-
sure by using Bernoulli’s theorem and the Hagen-Poiseille
equation [5] to estimate the mechanical contributions of
non-viscous flow and the viscosity correction during the
flow in the stomach and in the air, respectively.
2
z
2r
P0
P=P0+Pt
2R
FIG. 2: Model for penguin’s stomach. We assume that the
faeceses are prepared in a fictitious cylindrical tank with the
radius R. The depth zis estimated from the fluid volume
Vafter the firing. The pressure in the stomach is given by
P=P0+Ptwhere P0= 1013 hPa and Ptare the atmospheric
pressure and the rectal pressure, respectively.
This paper is organized as follows. In Sec. II, we ex-
plain our setup for the projectile motion of penguin’s
faeces and the evaculation of them from their intestines.
The latter is used for the estimation of penguin’s rec-
tal pressure. In Sec. III, we show Newton’s equation of
motion for the faeces after the shoot. We show the max-
imum flying distance at arbitrary angle and height. In
Sec. IV, we discuss effects of the viscous air resistance
during the projectile motion. In Sec. V, we calculate
the rectal pressure under the assumption that the faeces
liquid can approximately regarded as an ideal fluid. In
Sec. VI, we estimate the additional contribution to the
rectal pressure due to the viscosity correction within the
Hagen-Poiseuille equation. Finally, we summarize this
paper in Sec. VII.
II. SETUP
Before moving to the calculation, we explain the con-
figurations and the parameters we consider in this work.
Figure 1shows the situation where a Humboldt pen-
guin shoots the faeces out from the higher place with
the height h. Here, we employ an initial velocity |v0| ≡
v0= 2.0 m/s reported in Ref. [2] as a typical value. The
ejection angle is denoted by θ, and the gravitational con-
stant is fixed at g= 9.8 m/s2.
Figure 2shows the effective model of penguin’s stom-
ach. We employ properties of the faeces used in Ref. [2],
where the mass density ρ= 1141 kg/m3, the viscosity
η= 0.02 ∼0.08 Pa·s are employed. In our model the ra-
dius of bottom hole is given by r= 0.004 m. Although it
is known that penguin’s rectum has a form of a straight
tube [6], in this work we approximate it as a cylindrical
tank with the radius Rfor simplicity. The pressure in the
stomach is given by P=P0+Ptwhere P0= 1013 hPa
and Ptare the atmospheric pressure in the air and pen-
guin’s rectal pressure, respectively.
III. NEWTON’S EQUATION OF MOTION
WITHOUT AIR RESISTANCES
We consider Newton’s equations of motion
md2x
dt2= 0,(3.1)
md2y
dt2=−mg, (3.2)
where mis the total mass of penguin’s faeces. By solving
them, one can obtain textbook results
vx=dx
dt =v0x≡v0cos θ, (3.3)
vy=dy
dt =v0y−gt ≡v0sin θ−gt, (3.4)
x(t) = v0tcos θ, (3.5)
and
y(t) = h+v0tsin θ−1
2gt2.(3.6)
We obtain the grounding time tgfrom y(tg) = 0, which
reads
1
2gt2
g−v0ytg−h= 0.(3.7)
Thus, one can obtain
tg=v0sin θ
g+sv2
0sin2θ
g2+2h
g.(3.8)
Since the flying distance is given by d=v0tgcos θ, we
obtain
d=v0cos θ
v0sin θ
g+sv2
0sin2θ
g2+2h
g
.(3.9)
In Fig. 3, we plot das a function of θat a typical height
h= 1.1 m. The angle θm ax where dbecomes maximum
is given by the condition
∂d
∂θ θ=θmax
=−v0sin θ
v0sin θ
g+sv2
0sin2θ
g2+2h
g
+v0cos θ
v0cos θ
g+v2
0sin θcos θ
qv2
0sin2θ
g2+2h
g
= 0.
(3.10)
3
0
0.2
0.4
0.6
0.8
1
1.2
-90 -60 -30 030 60 90
θ [deg.]
d [m]
FIG. 3: The flying distance das a function of θat h= 1.1 m.
From Eq. (3.10) we obtain
θmax = sin−1
1
r21 + hg
v2
0
.(3.11)
In this regard, in terms of hand v0, the maximum flying
distance dmax is given by
dmax =v0cos
sin−1
1
r21 + hg
v2
0
×
v0
g
1
r21 + hg
v2
0
+v
u
u
t
v2
0
2g21 + hg
v2
0+2h
g
.
(3.12)
Figure 4shows h-dependence of dmax at θ=θmax . The
inset shows θmax as a function of h. In the case of h=
1.1 m which is typical height of rocks in the penguin’s
area of the Katsurahama aquarium, we obtain dmax =
1.03 m at θmax = 21.6◦. It is slightly longer the case
of the horizontal ejection d(θ= 0) = 0.95 m. We note
that the maximal height of rocks is h= 2.0 m in the
Katsurahama aquarium. In this case, we obtain dmax =
1.34 m at θmax = 16.9◦and d(θ= 0) = 1.24 m. Since
the air resistance in principle shortens das we discuss in
Sec. IV, this value can be regarded as an upper bound for
dmax. Therefore, we found that penguin keepers should
keep the distance being longer than 1.34 m from penguins
trying to eject faeces in the Katsurahama aquarium.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
00.5 11.5 2
0
15
30
45
60
0 0.5 11.5 2
θmax [deg.]
h [m]
dmax [m]
h [m]
FIG. 4: The maximum flying distance dmax as a function of
h. The inset shows the corresponding angle θmax .
IV. NEWTON’S EQUATION OF MOTION
WITH VISCOUS RESISTANCE
Here, we discuss effects of the air resistance. For sim-
plicity, we consider the case with viscous resistance which
is proportional to v. The equations of motion are given
by
md2x
dt2=−kvx,(4.1)
md2y
dt2=−mg −kvy.(4.2)
Resulting velocities and distances are given by
vx(t) = v0xe−k
mt,(4.3)
vy(t) = v0ye−k
mt+mg
k(e−k
mt−1),(4.4)
x(t) = mv0y
k(1 −e−k
mt),(4.5)
and
y(t) = h+m
kv0y+mg
k(1 −e−k
mt)−mg
kt. (4.6)
We note that by expanding them with respect to k, one
can reproduce the results in the absence of the resistance.
Since vx(t) exponentially decreases due to the nonzero k,
one can confirm that dmmax with k= 0 gives an upper
bound for the distance. We note that in this case the
grounding time is given by
tg=m
kW0−1 + kv0y
mg e−1+ kv0y
mg +k2h
m2g
+m
k+hk
mg +v0y
g,(4.7)
4
where W0(x) is the main branch of the Lambert Wfunc-
tion. The distance can be obtained by d=x(tg).
We also note that the resistance given by k|v|vis more
realistic in the air [7]. In such a case, an approximated
solution for a low-angle tra jectory has also been obtained
by using the Lambert Wfunction [8, 9]. We note that
an analytic solution in this case can be obtained by the
homotopy analysis method [10]. In addition, the viscosity
of faeces itself may shorten the flying distance due to the
energy dissipation originating from the internal friction.
More sophisticated treatments are required to take such
an effect into account. However, our estimation for the
upper bound on dmax is robust against these resistance
effects.
V. BERNOULLI’S THEOREM AND
ABDOMINAL PRESSURE OF PENGUINS
Bernoulli’s theorem for non-viscous faeces liquids is
given by
1
2ρv2+ρgz +P=const. (5.1)
where zis the initial height of liquids in a penguin as
shown in Fig. 2. While the faeceses are immobile un-
der the abdominal pressure Painitially, they are released
with v0in the air. Such an assumption gives
ρgz +Pa+P0=1
2ρv2
0+P0,(5.2)
where P0is the atmospheric pressure. At this stage, we
do not consider the viscosity correction. Thus we obtain
Pa=1
2ρv2
0−ρgz. (5.3)
The unknown parameter zcan be estimated from
z=V
πR2,(5.4)
where Vis the fluid volume after the ejection. Ris the
radius of stomach. For simplicity, we use R= 0.1 m.
Here, one may notice that the system is quite similar
to the so-called tank orifice with a vena contracta [12,
13]. In such a case, the cross-section area of non-viscous
flow after the ejection is approximately given by Cπr2,
where C≃0.611 is a typical value of the coefficient of
contraction [12]. In this regard, we obtain
V=Cπr2ℓ
=Cπr2Ztg
0
dtqv2
0x+ (v0y−gt)2
=Cπr2
2g"v0yv0−(v0y−gtg)q(gtg−v0y)2+v2
0x
−v2
0xln p(gtg−v0y)2+v2
0x−gtg+v0y
v0+v0y!#,
(5.5)
where ℓis the total path of the faeces. In particular, in
the case of θ= 0 (namely, v0y= 0 and v0x=v0), we
obtain
V=Cπr2
2g"gtgqg2t2
g+v2
0
−v2
0ln
qg2t2
g+v2
0−gtg
v0
.(5.6)
By substituting Eq. (5.5) to
Pa=1
2ρv2
0−ρg
πR2V, (5.7)
we can obtain Pa≃2.3 kPa at θ= 0 and h= 0.2 m
following the previous work [2]. It is smaller than the
estimated pressure 4.6 kPa in Ref. [2] where the initial
pressure for the acceleration was obtained by Pa=ρv2
0.
However, Paare small compared to the viscosity effect
which is addressed in Sec. VI.
VI. HAGEN-POISEUILLE EQUATION AND
VISCOSITY EFFECT
The Hagen-Poiseuille equation gives a relation between
an additional pressure and fluid flow rate Qfor laminar
flow though a cylindrical pipe [11] as
Q=πr4α
8η,(6.1)
where α=∇Pband Qcan approximately be given by
α≃Pb
ℓand Q≃V
tg, respectively. Therefore, we obtain
Pb=8V ℓη
tgπr4,(6.2)
which is consistent with Ref. [2]. Furthermore, we esti-
mate the additional pressure contributions for the rectal
flow as
Pc=8V zη
tdπR4≃4V v0η
πR4,(6.3)
where tdis the flowing time inside the intestine. We have
estimated it as td≃2z
v0≃0.45 ms. Even the maximum
duration tg+td≃0.20 s is quite short compared to
the urination time 8.2(M/kg)0.13 s obtained in Ref. [3]
(where Mis the mass of an animal). We note that a
typical value of Mfor a Hunboldt penguin is about 4
kg. In this way, the total rectal pressure is given by
Pt=Pa+Pb+Pc.
Figure 5shows the viscosity dependence of Ptwith
r= 0.004 m. For comparison, we plot the previous result
in Ref. [2] and our result of Ptwithout C. While the
previous work gives Pt= 8.56 ∼20.6 kPa for η= 0.02 ∼
0.08 Pa·s (noting that ηmay change due to the physical
5
100
101
102
103
104
105
106
η [Pa･s]
0.001 0.01 0.1 110 100
This work (with C)
Ref. [2]
P0
Pt [Pa]
r = 0.004 m
This work (w/o C)
FIG. 5: Calculated penguin’s rectal pressures Ptwith and
without the coefficient of the contraction C, as functions of
the viscosity η. The dash-dotted curve represents the result
of Ref. [2]. For comparison, we also show P0.
condition of penguins), our results show Pt= 8.75 ∼28.2
kPa, which is up to 1.4 times of the previous work. This
is because we consider a longer curved path ℓof flying
faeces to estimate Vwhereas only the horizontal distance
was taken account in Ref. [2]. We note that if we neglect
the contraction as C= 1, our result gives larger values
Pt= 12.9∼44.7 kPa since Vbecomes larger for given
r. These results are also larger than the bladder pressure
Pbladder = 5.2(M/kg)−0.01 kPa in Ref [3]. This difference
may originate from the physical configurations. Whereas
we consider the projectile ejection of faeces, in Ref. [3] a
vertical motion of urine was examined. In addition, since
penguins possess the cloacas which combine the digestive
and urogenital tracts [14], the visocity and density of
penguin’s faeces (actually mixing with urine) should be
inhomogeneous in the current case. The effect of such an
inhomogeneity is left for future work. We note that while
we use rough approximations for Rto calculate Paand
Pc, these contributions are found to be small compared
to Pb.
In the end of this section, to realize how strong the es-
timated penguin’s rectal pressure is, let us demonstrate
how far a liquid-like object blasted off from a human be-
ing with a severe stomachache flies if his/her rectal pres-
sure is as strong as penguin’s one. Here, we assume that
this liquid behaves as a nearly perfect fluid [15] and its
density is same as water ρ= 103kg/m3, for simplicity.
In this case, we obtain v0= 7.51 m/s from v0≃p2Pt/ρ
with Pt= 28.2 kPa. If the liquid is horizontally launched
from the hip with h= 0.85 m (which is a typical length of
legs), the flying distance can be estimated as d= 3.13 m,
which greatly excesses the typical height of a human be-
ing (1.7 m) we mentioned in the introduction. He/she
should not use usual rest rooms. Although we assume an
ideal situation to obtain the numerical value, one can eas-
ily understand the incredible power of penguin’s rectum
in this way.
VII. SUMMARY
To summarize, we have discussed a projectile tra jec-
tory of penguin’s faeces induced by the strong rectal pres-
sure. We have estimated the upper bound for the max-
imum flying distance dmax = 1.34 m for the case with a
typical initial velocity v0= 2.0 m/s of faeces, from the
higher place with the height h= 2.0 m. In the presence
of the air resistance which is proportional to the velocity,
we show that the grounding time and the flying distance
of penguin’s faeces can be expressed in terms of the Lam-
bert Wfunction.
Furthermore, based on the hydrodynamic approach,
we have revisited penguin’s rectal pressure with an ap-
proximation combining the Bernoulli’s theorem and the
Hagen-Poiseulle equation. The obtained rectal pressure
is larger than the previous estimation. The main differ-
ence originates from the treatment of faeces fluid volume
flying in the air. Our results would contribute to further
understanding of ecological properties of penguins.
In this paper, we have used simplified equations of
motion to track the trajectory of faeces. To obtain a
more quantitative results, it is necessary to solve hydro-
dynamic equations of faeces in the air and in the stom-
ach, which are left for future work. More sophisticated
treatment for the vena contracta would be important. It
is also interesting how complex hydrodynamic behaviors
such as turbulence appear during the ejection of faeces.
We expect that our results would also be a useful ex-
ample for teaching the classical mechanics in the under-
graduate course. One can recognize that fundamental
physics and mathematics we learn in schools describe in-
teresting aspects of various things surrounding our daily
life, as demonstrated by T. Terada [16].
Acknowledgments
The authors thank K. Iida, T. Hatsuda, A. Nakamura,
and Y. Kondo for useful discussion and the Katsurahama
aquarium for the supports and giving us an opportunity
to observe Humboldt penguins.
[1] P. B. Borboroglu, and P. D. Boersma, Penguins: Natu-
ral History and Conservation. (University of Washington
Press, 2013).
[2] V. B. Meyer-Rochow and J. Gal, Polar Biol. 27, 56
6
(2003).
[3] P. J. Yang, J. Pham, J. Choo, and D. L. Hu, Proc. Natl.
Acad. Sci. USA 111, 11932 (2014).
[4] P. J. Yang, M. LaMarca, C. Kaminski, D. I. Chu, and D.
L. Hu, Soft Matter 13, 4960 (2017).
[5] J. N. Newman, Marine Hydrodynamics, Cambridge,
USA, MIT Press (1977).
[6] A. S. King and J. McLelland, Birds: their structure and
function. Bailliere Tindall, London, pp 310311 (1984).
[7] G.W. Parker, Am. J. Phys. 45, 606, (1977).
[8] C. H. Belgacem, Eur. J. Phys. 35, 055025 (2014).
[9] C. H. Belgacem, Journal of Taibah University for Science,
11, 328 (2017).
[10] K. Yabushita, M. Yamashita, and K. Tsuboi, J. Phys. A:
Math. Theor. 40, 8403 (2007).
[11] C. Truesdell and K. R. Rajagopal, An Introduction to the
Mechanics of Fluids. Birkhuser, Boston (2009).
[12] J. H. Lienhard and J. H. Lienhard J. Fluids Eng. 10, 422
(1984).
[13] A. Hicks and W. Slaton, Phys. Teach. 52, 43 (2014).
[14] C. A. Oliveira, R. M. Silva, M. M. Santos, and G. A. B.
Mahecha, J. Morphol. 260, 234 (2004).
[15] P. K. Kovtun, D. T. Son, and A. O. Starinets, Phys. Rev.
Lett. 94, 111601 (2005).
[16] M. Matsushita, Evol. Inst. Econ. Rev. 6, 337 (2010).
