The Physics Teacher • Vol. 53, March 2015 (169 - 173)
Water Jets from Bottles, Buckets, Barrels,
and Vases with Holes
Vjera Lopac
University of Zagreb, Zagreb, Croatia
Observation of the water jets flowing from three equidistant holes on the side of a
vertical cylindrical bottle is an interesting and widely used didactical experiment
illustrating the laws of fluids in motion. In this paper we analyze theoretically and
numerically the ranges of the stationary water jets flowing from various rotationally
symmetric vessels with holes, in dependence on the height of the holes above the bottom,
on thickness of the block supporting the vessel and on different shapes of the vessel
profile. This investigation was motivated by controversial descriptions and illustrations
repeatedly found in the physics textbooks and by the fact that previously in the physics
teaching literature only the cylindrical vessel was treated.
The level of the fluid is supposed to be constant and the openings extremely
small. In further text we will continue referring to water, but all conclusions will be
applicable to any incompressible and inviscid fluid of density  in the homogenous
gravitational field g.
According to the Bernoulli equation, for a cylindrical vessel filled to the level H =
const. and for the hole at the height h above the bottom, the velocity of efflux is given by
Torricelli’s law
v  2g H  h .
(1)
The range of the water jet at the bottom level is
D  2 hH  h .
(2)
The longest range has the jet coming from the opening in the middle of the bottle,
h = H/2. The ranges of jets flowing from two holes placed at distances H/4 below and
above the central hole are shorter, but equal to each other. However, in many books and
papers erroneous illustrations can be found, ascribing the longest range to the lowest jet
and using it as an experimental proof of the fact that the hydrostatic pressure is highest
near the bottom of the bottle.
The error has been noticed and was thoroughly discussed in the didactical
literature. In years before 1989, a number of papers containing the critical reviews1-5 of
the erroneous results were published. They were focused on the meaning and significance
of the correct result1,2,4-6. The series of notes4-8 was ended with the editor's conclusion:
"The correspondence on water jets is now closed." However, publications with erroneous
description of experiments continued to appear. Confusing illustrations could be found
even in some globally popular books9,10. Several extensive reports followed11-13, in which
authors drew attention to the crucial role of the supporting block on which in most
experiments the vessel was placed1,5,7,8. It was noticed that its height B strongly
influences the range of the water jet13.
In Section 1 of this paper we evaluate two general equations. The first of them
describes the shape of the parabolic jet, the second determines the jet range in
dependence on the distance of the hole from the bottom and on thickness of the support,
for various profiles of rotationally symmetric vessels. In Section 2 we apply these
equations to the cylindrical bottle. In Section 3 several other vessel shapes are analyzed.
In Section 4 results are summarized and discussed with respect to their didactical
potential and historical context.
1. General equations for shapes and ranges of the water jets
We consider the water jets flowing from the orifice placed at the height h above
the base level of the vessel. The vessel is filled with water up to the level H which is
maintained fixed. The vessel stands on the cylindrical block of height B. The reference
system for describing the jet equation is chosen so that the x-axis coincides with the
horizontal base level of the support, and the y-axis with the vertical wall of the support
(Fig. 1). The shape of the vessel is described by the function S(h) for 0  h  H . Several
shapes will be considered, all depending on a positive parameter L, which denotes the
difference between the largest and the smallest radius of the vessel. The tangent to the
profile curve closes an angle with the vertical axis. One introduces a new shape
parameter
dS
a  tan  
.
(3)
dh
The water jet exits perpendicularly to the wall of the vessel. The angle  can be positive
or negative. If   0 , the water flows downward. If < 0 it flows upward and the
parabolic jet exhibits a typical maximum (Fig. 1).
Fig. 1. Meaning of variables h, S, , x, y, D, B, H and L.
2
The velocity v is given by Eq. (1) and its horizontal and vertical components are,
respectively,
and v y  vsin  .
(4)
vx  vcos
Using the symbol t for time, the coordinates of a point on the parabola, obtained from Eq.
(4), are
1
y  B  h   v y t  gt 2 .
(5)
x  S  v x t and
2
By combining Eqs. (4) and (5), one obtains the equation of the parabolic shape of the jet:
y  B  h   a  x  S  
1  a2
x  S 2 .
4H  h
(6)
With y  0 , one obtains the equation for the jet range x  D in dependence on h:
D  S 2 1  a 2   4aH  h D  S   4B  h H  h   0 .
(7)
Generally this equation should be solved numerically. However, for smooth shapes
investigated in this paper the range D can be obtained by solving Eq. (7) as a quadratic
equation in D  S  and taking its larger root, which gives
DS


2a
H  h   2 2 a 2 H  h 2  1  a 2 H  h B  h  .
2
1 a
1 a
(8)
2. Cylindrical bottle with holes
Shapes and ranges of the water jets
Two carefully prepared experimental arrangements, one with the supporting block and
the other without it, are shown in Figs. 2 and 3. For the cylindrical bottle S  0 ,   0 ,
a  0 and L  0 . The shape parameters are listed in row A of Table I.
Equation (2) for B = 0 is obtained from Eq. (8). The calculated jets from five
holes with h/H = 0.01, 0.25, 0.50, 0.75 and 0.99 on the walls of a cylindrical bottle are
shown in Fig. 4(a). Horizontal lines in the picture facilitate the comparison of water jet
ranges for four different thicknesses B of the support. The calculated values of the jet
range in dependence on h for different B are compared in Fig. 4(b). The lowest curve (for
B = 0) is symmetrical and was already depicted in Refs. 1 and 2.
The situation where the range D increases with increasing depth H-h and where
the lowest jet has the largest range is possible only if the bottle is placed on a support of
sufficient thickness. For the thickness B of the supporting block the range of the water jet
resulting from Eq. (8) is13
D  2 B  hH  h .
(9)
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If D is not allowed to decrease when h decreases, there must be dD / dh  0 ; hence
 B  H  2h  0 or B  H  2h . As this must be valid for any possible h (thus also for
hmin= 0), one concludes that the lowest jet has the largest range only if
BH,
(10)
i.e., if the thickness B of the supporting block is equal to or larger than the distance H
from the base level of the vessel to the free surface.
Table I. Profile equations for considered vessel shapes
A
B
C
D
E
F
Vessel shape
Vessel profile S (h)
Cylindrical
bottle
Bucket
S 0
Slope of the vessel
profile a  tan 
a  tan  = 0
L
h
H
L
S  h
H
4L
S  2 h H  h 
H
L
>0
H
L
a  tan    <0
H
4L
a  2 H  2h 
H
L 
h
sin  2 
2 
H
L
 hN 
S
sin  2

2N 
H 
L
h

cos 2 
H
H

L
hN 

a
cos 2

H
H 

Conical
bottle
Barrel with
parabolic
profile
Sinusoidal
vase
Corrugated
vase N > 1
S
S
Fig. 2. Water jets from three holes on a
cylindrical bottle placed on the supporting
block.14
a  tan 
a
Fig. 3. Water jets from a bottle with three
holes without the supporting block12. The
longest range has the jet flowing from the
hole in the middle of the bottle. (Set up and
photo by Adrian Corona Cruz.)
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Dependence of the jet range on pressure and energy
Since the pressure p  g H  h of the fluid depends on its density  and
increases with the increasing depth H–h of the hole, Eq. (2) for the range of the water jet
can be rewritten as
D
2
g
p pmax  p  ,
(11)
where pmax  gH . This shows that for small h the range increases, but for larger h
decreases with pressure.
The jet range can also be expressed by means of the kinetic energy Ek and the
gravitational potential energy E p of a small part of the fluid of mass m placed at the
orifice at the height h above the bottom. Since Ek  mv 2 / 2  mgH  h and,
measured from the base level of the support, Ep  mg( B  h ) , Eq. (11) for the jet range
can be written as
2
(12)
D
Ek Ep .
mg
3. Water jet ranges for different vessel shapes
Bucket with holes
The bucket is a truncated conical vessel wider at the top. The experiment with the
leaking bucket was suggested in Ref. 10, while Fig. 5. illustrates how easily such system
can be realized in everyday life. The shape profile of the vessel is a tilted straight line,
and the angle is constant and positive. Their dependence on h is shown in the row B of
Table I. Parabolic jets calculated from Eq. (6) for different B, a = tan and L/H =
0.2, are shown in Fig. 6(a) and their ranges in dependence on h are shown in Fig. 6(b).
Similar results and graphs hold for a conical bottle with holes.
Barrel with holes
Barrels are traditional vessels often seen in real life. An example is shown in Fig.
7. We consider here the barrel with a parabolic profile, specified in row D of Table I.
Shapes of the parabolic jets and the range dependence on h for a barrel are shown in Figs.
8(a) and 8(b).
Vases with holes
A vase is a vessel with sinuous, in our case also a sinusoidal, profile. The profile
and its slope are specified in the row E of Table I. The range curves for the vase exhibit
oscillations that are obviously caused by the periodical nature of the shape profile. If the
vessel profile is folded more than once, the shape and its slope can be generalized to
equations given in row F of Table I, with values N > 1. Results are the corrugated vase
shapes and somewhat surprising oscillating curves depicting the dependence of the jet
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range on h. In Figs. 9(a) and 9(b) the calculated results are shown for the corrugated vase
with N = 8. If the vessel profile is folded more than once, the shape and its slope can be
generalized to equations given in the row F of Table I, with values N > 1. Results are the
corrugated vase shapes and somewhat surprising oscillating curves depicting the
dependence of the jet range on h. In Figs. 9(a) and 9(b) the calculated results are shown
for the corrugated vase with N = 8.
Fig. 4. (a) Water jets from holes on the
cylindrical bottle.
Fig. 4. (b) Dependence of the water jet
range on h for the cylindrical bottle.
Fig. 5. Jets flowing from the bucket with holes.15
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Fig. 6. (a) Water jets from holes on the
walls of a bucket.
Fig. 6. (b) Dependence of the water jet
range on h for the bucket.
Fig. 7. Barrel with a hole in the middle: from it the water jet exits horizontally.17
Fig. 8 (a) Water jets from holes on the
walls of a barrel.
Fig. 8. (b) Ranges of the water jets from
holes on the walls of a barrel.
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Fig. 9. (a) Water jets from holes on the Fig. 9.(b)Dependence of the water jet range
corrugated vase with N = 8
on h for the corrugated vase with N = 8
4. Discussion and conclusions
Discussion of the calculated results
We have investigated theoretically and numerically the water jets flowing from vessels of
various shapes with holes. Interesting results were obtained for different profile shapes
and support thicknesses. Comparing the jet range curves for various shapes one finds the
following.
a) For the highest jet (h/H = 1) the range is D = 0 for all examined shapes, except for the
conical bottle for which it is negative; actually, it would be negative if the jet would not
be stopped by the vessel itself.
b) The lowest jet (h = 0) has the range D = 0, provided that there is no supporting block
below the vessel, again for all considered shapes except for the conical bottle, for which
all jets, including the lowest one, flow upward.
c) There are three categories of range curves: those decreasing with h, those with one
maximum for a special value of h, and the complicated curves with many maxima and
minima reflecting the shape oscillations of the vessel profile.
d) The derivation can be extended to many other vessel shapes, including the vessels with
rough or uneven walls suggested in Ref. 17, where also the effects of deterministic chaos
could be expected.
Didactical possibilities of the obtained results
We have in mind several possible applications of the obtained results in the teaching
process.
a) The theoretical calculations of the water jet shapes and dependence of their ranges on h
do not lie beyond the usual mathematical and programming skills of high school and
college students. They could be used as attractive numerical tasks for the physics class.
b) The shapes encountered in such investigation can be applied, in the sense of ideas
proposed in Ref. 18, to connect the subjects of physics with various geometrical objects.
One can consider different parabolic shapes of the jets and compare them with the paths
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observed in the projectile motion. Various vessel profiles can be presented as examples of
interrelationship between the geometry and the motion of flows and particles.
c) Another possible use is experimental: the vessels of proposed shapes, found in the
environment or custom made, can be used to observe the ranges of the water jets and
compare them with results obtained by calculation.
Besides rotationally symmetrical vessels, which are appropriate for visualizing the shapes
of the water jets, we propose a different type of vessel that would make possible the
visualization of the range curves in dependence on h. It could be produced in the
laboratory or ordered from specialized shops. For the vertical wall, such vessel would
have the form of a rectangular block shown in Fig. 10. The jets exiting from a series of
holes made along the diagonal of the longest wall would reach the bottom level at
different distances. They could leave traces depicting the lowest curve shown in Fig. 4(b).
The same type of vessel could be produced with different profiles S(h), and the support
could be added when needed.
d) Finally, the subject offers a large choice of historical issues. There is the story of
Leonardo da Vinci, who first made drawings of the jets flowing from cylindrical
containers with holes11,13– albeit erroneously – in the 15th century, 130 years before
Torricelli. The importance of different shapes can also be emphasized by quoting
Johannes Kepler, who spent a lot of his time calculating the volumes and shapes of
barrels19.
Acknowledgments
The author wishes to thank V. Volovšek for useful comments and J. Slisko for permission
to use the photo from Ref. 12.
Fig. 10. The vessel shape proposed for the experimental investigation of the jet ranges.
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References
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14. http://www.kids-fun-science.com/images/sa2-kids-science-activities.jpg
15. http://commonscold.typepad.com/.a/6a00d8345280a669e201348053709a970c-popup.
16. Photo: V. Lopac
17. "Investigating the pressure of a water column“;. Nuffield Foundation, Practical
Physics, http://www.nuffieldfoundation.org/practical-physics/investigating-pressurewater-column
18. Thomas B. Greenslade Jr., "The shapes of physics", Phys. Teach. 51, 524–534 (Dec.
2013).
19. Roberto Cardil (MatematicasVisuales), "Kepler: The Volume of a Wine Barrel Introduction," Loci (January 2012).
http://www.matematicasvisuales.com/english/html/history/kepler/keplerbarrel.html
Vjera Lopac is Professor of Physics at the University of Zagreb, Croatia. Her research
interests include nuclear physics, nonlinear physics, numerical simulation, educational
physics, metrology in physics and scientific terminology. She is author of several physics
textbooks and translator of books on popularization of science.
vlopac@fkit.hr
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