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On the trajectories of projectiles depicted in early ballistic woodcuts
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IOP PUBLISHING EUROPEANJOURNAL OFPHYSICS
Eur. J. Phys.33 (2012) 149166 doi:10.1088/0143-0807/33/1/013
On the trajectories of projectilesdepicted in early ballistic woodcuts
Se an M Stewart
Department of Mathematics, The Petroleum Institute, PO Box 2533, Abu Dhabi, United ArabEmirates
E-mail:sstewart@pi.ac.ae
Received 9 August 2011, in final form 21 October 2011Published 29 November 2011
Online atstacks.iop.org/EJP/33/149
Abstract
Motivated by quaint woodcut depictions often found in many late 16th and17th century ballistic manuals of cannonballs fired in air, a comparison oftheir shapes with those calculated for the classic case of a projectile movingin a linear resisting medium is made. In considering the asymmetrical natureof such trajectories, the initial launch angle resulting in the greatest forwardhorizontal skew of the trajectory is found and compared to the correspondinglaunch angle which maximizes the range of the projectile. Limiting behaviourin the shape of the trajectory in the strong damping limit is also considered andcompared to peculiar straight-lined triangular trajectory woodcuts from the late16th century.
(Some figures may appear in colour only in the online journal)
1. Introduction
The classic problem of the motion of a projectile in a resisting medium continues to be the
subject of ongoing interest [127]. Part of this interest stems from the problems ability to
throw up a wealth of interesting questions suitable for investigation by intermediate-level
undergraduate students of applied mathematics and physical sciences. Questions concerning
the shape of a projectiles trajectory in air are instructive given the historically important role
they played in the development of early modern classical mechanics. The rise of military
ballistics, particularly cannon fire, and their growing use in Europe throughout the 15th and16th centuries demanded a greater mathematical understanding of the projectile problem
beyond that which was merely descriptive. To be effective in warfare, the need to accurately
predict the range of cannon fire arose and meant a quantitative understanding of the precise
nature of a projectiles trajectory as it moved through air.
This paper is motivated by a recent article by La Rocca and Riggi [23], who questioned
whether the trajectories of cannon fire depicted in the woodcuts of many gunnery or
military ballistic manuals prior to the 18th century could be correctly reproduced from
0143-0807/12/010149+18$33.00 c 2012 IOP Publishing Ltd Printed in the UK & the USA 149
http://dx.doi.org/10.1088/0143-0807/33/1/013mailto:sstewart@pi.ac.aehttp://stacks.iop.org/EJP/33/149http://stacks.iop.org/EJP/33/149mailto:sstewart@pi.ac.aehttp://dx.doi.org/10.1088/0143-0807/33/1/0135/26/2018 Trayectoria de Proyectiles
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150 S M Stewart
Figure 1.A 1684 example typical of those prior to 1700 showing the trajectories of cannon fire in
air.
modern day Newtonian mechanics. Their answer, surprisingly, was almost. On looking
at the trajectories depicted in many of these woodcuts one finds asymmetric trajectories
with prominently skewed appearances. An illustrative example from 1684 [28] is shown in
figure 1. With their long, almost straight-lined ascents followed by very short and abrupt
descents, in attempting to reproduce similar looking curves, a linear resisting model will be
used. While the linear resistive model is not expected to quantitatively reproduce the behaviour
of projectiles fired in air it remains a useful first approximation to make as it allows one to
discover to what extent can such a model reproduce at least qualitatively in appearance the
prominently skewed trajectory shapes found in many early ballistic woodcuts of cannonballs
fired in air. In analysing these highly skewed trajectories, it is natural to ask at what angle to
the horizontal should a projectile in such a resisting medium be launched for its trajectory to
take on its greatest forward skew along the horizontal. Such a question does not appear to have
been asked until quite recently [25,2931].
Up until the late mediaeval period the Aristotelian concept for the motion of a projectile
in air held sway. Here the motion was considered to consist of two parts. The first part, known
as the violent part, was the result of the initial impetus required to launch the projectile into the
air. During this time the object was thought to move in a straight line along the direction of its
initial projection. Only after the initial impetus was exhausted did the second part, known as
the natural motion, come into play. The motion was natural in the sense that it was the normal,unimpeded motion. In the case of a projectile moving in air the natural motion was a vertical
descent back towards the centre of the Earth. By the dawn of modern classical mechanics
in the mid-16th century a third so-called mixed motion had been added between the violent
and natural parts. The mixed motion accounted for the smoothness observed in the path of a
projectile and was thought to be a mixture between the violent and natural parts. It is these
three motions which are depicted in figure1.To account for the mixed motion mathematically,
it had to be represented by geometrical curves known at the time and so was conceptualized, in
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Linearly resisted projectile trajectories 151
most instances, as an arc of a circle [32]. Such an idea is to be found in the first comprehensive
study of the motion of a cannonball fired in air by Tartaglia 1 (14991557) in his 1537 work
Nova Scientia, and in the work of later contemporaries, particularly Thomas Digges (154695)
in hisPantometriaof 1571 (and the expanded edition of 1591).
A century after Tartaglia, Galileo Galilei (15641642) in his workDialogues ConcerningTwo New Sciences of 1638 established, by considering the motion of a projectile to consist
of a uniform component along the horizontal and a uniformly accelerated component along
the vertical, that the path of a projectile in the absence of any resisting force was parabolic.
Incorporating the effects of air resistance on the motion of a projectile, with its observed
asymmetrical trajectory, however, proved to be problematic. Thus the concept of tripartite
motion for a projectile moving through air persisted well into the early 18th century. It was not
until the work of Isaac Newton (16421727) in 1687 did we find, for the first time, the problem
treated in an effective mathematical manner. Of the three books of his Principia, the second
book is devoted largely to the study of motion through resistive media. It commences with
the problem of resisted projectile motion in air. Initially, Newton considered a resistive force
that varied in proportion to velocity. By doing so he was able to construct mathematically
the trajectory for the resisted projectile by, as he writes, [with] the help of a table oflogarithms [33]. While Newton gave no explicit expression for the trajectory, the curves
he presented clearly show the expected asymmetry found in trajectories resisted by air. The
need to conceptually divide the trajectory into three composite parts was at once condemned
to history. Despite this, the tripartite motion idea for resisted trajectories lingered on for some
time after the publication of Newtons Principia. Why it persisted probably had more to do
with the simple and appealing Euclidean geometric connection to straight lines and circles the
former idea had rather than to any strong sense of conviction that the trajectory of a projectile
in air was actually approximated by such curves.
In this work, after briefly reviewing in section2 the well-known solution of a projectile
in a linear resisting medium the initial launch angle that gives the greatest forward horizontal
skew in the trajectory is presented in section 3. A comparison between the resulting trajectories
formed at this angle, and at the angle which maximizes the range of the projectile, to those
typically found depicted in early arterial and military ballistic woodcuts is then made. In
section 4 the limiting behaviour in the shape of the projectile trajectory in highly
resistive media is analysed and compared to peculiar looking straight-lined triangular
trajectory woodcuts from the late 16th century. We conclude with a brief summary in
section5.
2. Linearly resisted projectile motion
Consider an object of mass m launched with initial speedv0 from the origin at an angle of
to the horizontal over level ground and in a gravitational field gwhich is constant. In a linear
resisting medium the retarding force acting upon the object is considered to be proportional
to its velocity and directed in a direction opposite to its motion. Physically, this case applieswhen the Reynolds number Re is small (Re < 1)[34]; the Reynolds number itself being a
dimensionless measure of the magnitude of the inertial forces acting on the object relative to
the viscous forces as it moves through a resisting medium. When Re < 1, it is clear that the
viscous forces dominate, with the inertial forces being much smaller by comparison.
1 Niccolo Fontana is today better known asTartagliathe stammerer, the stammer being the result of a horrific sabrewound sustained to his jaw in his youth.
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152 S M Stewart
In air, for very small objects moving with relatively low speeds, the linear drag model
is the classic first approximation made to the more general problem of a projectile moving
in a resistive medium. Granted, a 17th century cannonball fired in air is not exactly expected
to satisfy each of these conditions. However, in keeping with our present aim of trying to
reproduce at least qualitatively the broad features of those trajectory shapes found depicted inearly ballistic woodcuts, while at the same time retaining analyticity, the linear model will be
used.
From Newtons second law, we have for the horizontal and vertical components of the
motion[35]
x= x, (1)
y= y g, (2)respectively. Here,is a positive drag coefficient per unit mass ( > 0) and is assumed to
remain constant during the motion of the projectile, while the dots above each of the coordinate
variables(x,y)denote derivatives with respect to timet. In the absence of air resistance, the
unresisted case is recovered when=
0.
Solving the above two differential equations in the usual way, on applying the initial
conditions
x(0)=0, x=v0cos (3)
y(0)=0, y=v0sin , (4)yields the following well-known equations of motion:
x(t)= v0cos
[1 exp(t)], (5)
y(t)=
g
2+ v0sin
[1 exp(t)] gt
. (6)
The asymmetry introduced into the trajectory of a projectile moving through a linearresisting medium is most readily seen from the equation for its trajectory. When time is
eliminated from(5) and (6), one obtains for the equation of the trajectory [35]
y(x)=
g sec
v0+ tan
x+ g
2 ln
1 sec
v0x
. (7)
On inspecting (7) the mathematical construction for the trajectory of a projectile in a linear
resisting medium using, as Newton noted, nothing more than a table of logarithms is readily
apparent.
Compared to the unresisted trajectory which is parabolic in shape and symmetric about
its vertex, the path of a projectile in the linear resisting model becomes asymmetric due
to the presence of an additional parameter introduced through the damping coefficient .
Physically, the introduction of asymmetries into the trajectory paths can be explained by
the decreasing horizontal component of the velocity of the projectile as it moves througha resisting medium. It results in the trajectory being longer and shallower on ascent while
shorter and steeper on descent. The vertex in the trajectory therefore occurs closer to its
final impact point rather than its initial launch point; a feature in common agreement with
woodcut depictions for cannonballs fired in air found in old gunnery manuals. The forwardly
skewed appearance in the trajectory is a characteristic feature for all projectiles launched
in a linearly resisting medium and has been rigorously established elsewhere (see, for
example, [5,15]).
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Linearly resisted projectile trajectories 153
8
4
3
8
2
Figure 2. Normalised horizontal distance travelled to reach the trajectory peak versus initial launch
angle fora number of differentresistiveconstants(= 0, 0.1, 0.5, 1.0, 2.0, 5.0 s1, topto bottom).
3. Angle for the greatest forward horizontal skew
3.1. Optimal angle for the maximum forward skew
In order to find the launch angle which maximizes the forward horizontal skew for a
projectile in a linear resisting medium we begin by finding the coordinates for the vertex
of the trajectory in the Cartesian plane. At the trajectory peak, since the projectile is at its
greatest height above ground level, its corresponding speed in the vertical direction vanishes.
Thusy= 0 and it allows the time of ascent ta taken by the projectile to reach its peak to befound. Differentiating (6) with respect to time, on settingy=0 and solving fort, one readilyfinds
ta=1
ln
1 + v0
gsin
. (8)
Substituting(8) for the time taken for the projectile to reach its vertex into (5), the horizontal
distance travelled by the projectile on ascent to its peak will be
xa( )=v20sin 2
2g(1 + c sin ) , (9)
where the dimensionless quantity c
= v0/g has been introduced. Figure 2 shows a plot
of the horizontal distance travelled (these are normalized by dividing by v20 /2g so as toappear dimensionless) by a projectile to its peak in a linear resisting medium as a function
of the initial launch angle for the case ofv0= 10 m s1 and for various resistive constants(= 0, 0.1, 0.5, 1.0, 2.0, 5.0 s1, top to bottom). The broken curve is for the unresisted(= 0) case. From the figure it can be seen that the initial launch angle leading to the greatestforward horizontal skew moves progressively away from /4, the angle corresponding to the
greatest horizontal distance attained at the peak for an unresisted projectile towards smaller
angles as the resistance of the medium increases.
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154 S M Stewart
If we let the optimal angle of projection for the greatest forward skew be max, anexpression for this angle can be found. It will be the angle satisfying dxa/d = 0.Differentiating(9) with respect to and simplifying gives
dxa
d =v20 (1
2 sin2
c sin3 )
g(1 + c sin )2 . (10)Stationary points in(10) occur when the launch angle satisfies the cubic equation
cu3 + 2u2 1=0, (11)corresponding to when the numerator in (10) vanishes. Here we have set u=sin max.
The three roots to (11) are completely characterized by its discriminant [36], D =(27c232)/(108c4 ). Three different cases for the solution arise: (i) for three distinct real roots,D 0 corresponding toc >
32/27. In a linearly resisting medium
asc is positive Descartes Sign Rule [37] ensures that the maximum number of positive real
roots for (11) is at most one. Thus there will be only one angle max (0, /2)for whichxais maximized, a result clearly seen to be the case on inspecting figure 2. An expression forthe optimal angle is found on solving (11). While cubic equations can be solved for explicitly,
one is able to express their solution in a number of alternative but equivalent forms. Using the
so-called Viete form[36] for the solution of a cubic equation2, for the positive real root we
have
max=
sin1
4
3ccos
1
3cos1
27
16c2 1
2
3c
, 0< c
32
27
sin1
4
3ccosh
1
3cosh1
27
16c2 1
2
3c
, c>
32
27.
(12)
For weak damping, the expansion of the cosine term appearing in (12) in terms ofcabout
the origin is needed. It is
cos
13 cos1
2716 c2 1 = 12+ 328 c 332 c2 + 152512 c3 3128 c4 + . (13)So asc0+, if we write u=sin max and employ(13) then in this limit (12)becomes
limc0+
u= limc0+
1
2 c
8+ 5c
2
64
2 c
3
32+
= 1
2, (14)
or limc0+max= /4, the unresisted result as expected. Note that the unresisted result of /4 for the greatest forward horizontal skew is not only approached as the drag coefficient
per unit mass for the resistive medium becomes very small ( 0+) but also for projectileslaunched with very small initial speeds since c0+ whenv0 0. Physically, this is to beexpected since as the resistance of the medium to motion is taken to vary in proportion to its
velocity, the smaller the velocity, the less the motion of the projectile will be affected by the
medium.
An interesting connection between the angle max to the golden ratio is found for thespecial case ofc=1. Settingc equal to unity in(12) gives
max=sin1
1
, (15)
2 If the more familiar CardanoTartaglia formula for the solution of a cubic is used, an expression for the positive realroot in terms of nested radicals can be found[29]. However, when 0 < c 0. For a further discussion on this point see [ 30,31].
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Linearly resisted projectile trajectories 155
where = (1 +
5)/2 corresponds to the golden ratio. Recall that the golden ratio and its
reciprocal 1 = (
51)/2= 1 are the two roots to the quadratic equation u2u1=0.Such a result, while initially intriguing, is not intrinsic to the problem itself. As Essen and
Apazidis [38] point out any physical problem that leads to a quadratic equation with adjustable
parameters will always result in the golden ratio if the parameters are chosen so as to giveu2 u 1=0. In our case when the parameter c is set equal to unity, a simple factorizationof (11) leads to the required quadratic equation for an apparent golden ratio connection.
Interestingly, when the parameter c is adjusted to unity the initial launch speed required to
give the golden ratio connection is v0=g/, a speed equal to the terminal speed of an objectfalling vertically under gravity in a linear resisting medium [31].
3.2. Optimal angle for the maximum range
An obvious question to ask is how does the angle which gives the greatest forward horizontal
skew in the projectile trajectory at its peak, namely max, compare to the initial launch anglewhich maximizes the range of the projectile, max. Until recently a closed-form expression
formax
was simply not possible. However, if expressed in terms of the so-called Lambert W
function, explicit closed-form expressions for not only the optimal angle which maximizes
the range of the projectile but also its time of flight and range become possible [ 913,15].
An overview of the most important results relating to the Lambert W function needed here is
provided in appendixA.
To find an explicit expression for the optimal angle max, we first need to find the
corresponding expression for the range R of a projectile in a linear resisting medium. Setting
y=0 in(7) and rearranging terms one findsR( )= cos
a[1 exp(B( )R())], (16)
where B( ) = a sec + b tan , a = /v0, b= 2/g, with the range being obtainedas the solution to this transcendental equation. If we set u = B( )R( ) and note that
B( ) cos /a=(), where()=1 + c sin ,(16) can be transformed into the far simplerformu= (1 eu ). (17)
After rearranging algebraically,(17) can be rewritten as
(+ u) e+u = e, (18)an equation exactly in the form of (A.1), the defining equation for the Lambert W function.
Its solution in terms of W is
( ) + u=W0(() e() ). (19)The principal branch is chosen as it is the branch which gives the non-trivial solution for the
range. Finally, by recognizing B( )
= 2g()(()
1)/(v20sin 2), in terms of () a
closed-form expression for the range becomes
R( )= v20sin 2
2g
() + W0(() e() )
()(() 1)
. (20)
The angle max which maximizes the range of a projectile is found from the angle for
which dR/d= 0. Finding the angle that leads to an explicit expression for max is howevernot a simple case of differentiating (20) and setting the result equal to zero before solving
for . Instead, it requires a few subtle algebraic manoeuvres and we follow a procedure first
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156 S M Stewart
advanced by Groetsch and Cipra in [2]. Differentiating (16) with respect to and setting
R( )=0 one obtains0= sin
a(1 eB()R() ) + cos
aB( )R( ) eB()R() . (21)
It is however apparent from(16) that
1 eB()R() =a sec R(), (22)while the derivative ofB( )is
B( )=a sec tan + b sec2 = a sin + bcos2
. (23)
Substituting (22)and (23) into(21) and simplifying leads to the following expression for the
maximum rangeRmax at the optimal angle of projection:
Rmax=c cos max
a(sin max+ c). (24)
Here,c=b/a= v0/g. It immediately follows that
B(max )Rmax=c(1
+c sin max )
sin max+ c . (25)Substituting(24) and(25) back into (16), and settingu=sin max, we arrive at
u
u + c =exp
c + c2u
u + c
, (26)
a transcendental equation in terms of W.
Before solving(26) for the general case, the solution of which can only be written in a
closed form by the introduction of the Lambert W function, consider the special case ofc=1.Here(26) is no longer transcendental and the solution u= (e1)1 readily follows. Theangle which maximizes the range of the projectile when c=1 is therefore
max=sin1
1
e
1
, (27)
a result pre-dating the formal arrival of the Lambert W function [39].Returning to the solution of (26)for the general case ofc= 1, the general strategy is to
write (26) in the form of(A.1). If we multiply both sides of (26)by the factor(c2 1), afterrearranging terms one has
ue
u + cc2 1
eexp
ue
u + cc2 1
e
= c
2 1e
, (28)
which is now exactly in the form for the defining equation for W. Its solution is
ue
u + cc2 1
e=W0
c2 1
e
. (29)
Solving foru, settingmax=sin1 u, and on combining the result with (27) yields
max=
sin1 cW0 c
2
1
e
c2 1 W0
c21e
, c=1
sin1
1
e 1
, c=1.
(30)
The choice of the principal branch is made on the following basis. Forc > 1, the argument for
the Lambert W function appearing in (30)is greater than zero and accordingly the principal
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Linearly resisted projectile trajectories 157
branch prevails. For 0< c < 1, the argument falls between 1/eand zero, making the choicebetween either branch possible. The secondary real branch is however rejected since it leads
to a negative value formax, a physical impossibility sincemax (0, /2).For the Lambert W function term appearing in (30), its Maclaurin series is given by
W0
c2 1
e
= 1 +
2c 2
3c2 + 11
236
c3 + . (31)
From (30) as c0+, if we write u=sin max and employ (31), one has
limc0+
u= limc0+
1
2c + 23
c2 + 2 5
3c + 11
36
2c2 +
= 12
, (32)
or limc0+max= /4, the unresisted result as expected.Finally, an expression for the maximum range at the optimal launch angle in terms of the
Lambert W function can be found. From simple trigonometry, an expression for the cosine at
the optimal launch angle follows from(30). It is
cos max=
(c2 1)2
2
+W0 c21e (c2
1)W0 c21e c2 1 W0
c21
e
, c=1
e2 2ee 1 , c=1.
(33)
Substituting(30) and(33) into(24) and simplifying yields
Rmax=
v20
g
c2
W0
c21e
+ 1
2c2(c2 1) , c=1
v20
g
1 2
e, c=1.
(34)
3.3. Comparison between the two optimal angles
A plot of the optimal angle for the greatest forward skew in the trajectory max (solid line),together with the optimal angle that maximizes the range max (broken line), as a function
of the dimensionless drag parameter c= v0/gis shown in figure 3. The optimal angles inboth cases are always less than /4. The curves suggest that each optimal angle decreases
monotonically asc increases, an observation to be established shortly for both angles, and as
has been noted earlier, each takes on their maximum value of /4 at c=0. Finally, and mostinterestingly, we see thatmax > max for allc > 0.
The final observation has particular consequences regarding the shape of the projectiles
trajectory. SinceR(max ) > R(max )but xa(max ) < xa(
max )we see for fixed initial launch
speeds the horizontal distance between the peak and its final impact point is less for a projectile
launched at an initial angle corresponding to the greatest forward horizontal skew compared tothe projectile launched at an initial angle which maximizes its range. Steeper vertical descents
are therefore achieved in projectiles launched at an initial angle equal to max compared tothose launched at an angle which maximizes the range. So projectiles launched at or very
near to those angles giving the greatest forward horizontal skew result, at least outwardly in
appearance, in trajectories resembling most closely those depicted in early gunnery or military
ballistics woodcuts. As an example, two trajectories resulting from a projectile launched at
each of the optimal angles max and max are given in figure4.
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158 S M Stewart
16
8
3
16
4
c = v0/g
Figure 3. Optimal angle of projection for the greatest forward skew (solidline) in thetrajectory of a
projectile in a linearresisting mediumas a function of thedimensionless drag parameter c= v0/gcompared to the corresponding angle which maximizes the range of the projectile (broken line).
Figure 4. Two different trajectories for a projectile fired with the same initial speed (v0 =300 m s1) in a linear resisting medium (= 1.0 s1) at launch angles equal to (i) the optimalangle for the greatest forward horizontal skew (solid line) and (ii) the optimal angle for the greatest
range (broken line). In appearance, the trajectory for the greatest forward horizontal skew with
its longer straight-lined ascent followed by its steeper and shorter vertical descent resembles mostclosely those trajectories found depicted in early ballistic woodcuts. To help aid the eye a short
vertical line segment has been drawn through the apex of each trajectory.
Historically, when applied to the art of warfare, cannon shot fired at initial angles resulting
in steep vertical descents caused minimal damage to besieged towns since on impact the
cannonball tended to do little else other than bury itself deep into the ground [40]. So while
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Linearly resisted projectile trajectories 159
early ballistic woodcut depictions of cannon fire were no doubt heavily influenced by the
prevailing notions held about motion at the time, in terms of military effectiveness a launch
angle resulting in steep vertical descents was best avoided if the intention was to inflict the
greatest possible damage with each shot.
To show max is a decreasing function ofc, wewrite u=sin maxand begin by consideringthe branch ofufor which 0 < c
32/27 in (12). Differentiating with respect toc gives
du
dc= 2
3c2
32 27c2f(c), (35)
where f(c)= 4
3c sin (c)
32 27c2(2 cos (c) 1) and (c)= cos1(27c2/161)/3. From (35) we see u will be a decreasing function of c provided f(c) < 0 for all
0 < c
32/27in (12) can alsobe shown to bea decreasing
function on its interval and accordingly will not be given here. Thus,uis a decreasing function
for all c > 0 and since the inverse sine function decreases monotonically as its argument
decreases,max also decreases as a function of the dimensionless drag parameter c. Finally, as
1
21 for c>
32
27, (37)
max is positive for all c > 0, and along with its decreasing, limiting behaviour provides thebound
0< max 0, we employ an approach which is a slight variation on what has been
used in the past [10,11]. We begin by rewriting (30)in the equivalent form of
max=sin1
c
exp(W0[(c2 1)/e] + 1) 1
. (39)
Here (A.1) has been used and we note that(39) is valid for all positive c includingc=1. Aswas done previously, it is convenient to write u= sin max and consider the behaviour ofu.Differentiating gives
du
dc= c
2 exp(W0( ) + 1) + W0( ) + 2[exp(W0 ( ) + 1) 1]2(W0( ) + 1)
, (40)
where we have set (c)=(c2 1)/eand the resultW0( ) e
W0 ( )+1 =c2 1 (41)
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160 S M Stewart
has been used and follows from(A.1). The denominator of(40) is clearly positive since the
principal branch for the Lambert W function W0(x) >1 for allxin its domain. To show thenumerator of (40) is also positive, let
h(c)=
c2
exp(W0( )
+1)
+W0 ( )
+2, (42)
then
h(c)= 2cW0( ) + 1
[W0( ) + exp(W0( ) 1)]. (43)
Sincec > 0, clearlyc2 1>1 orW0( ) e
W0 ( )+1 >1, (44)where (41) has been used. As the exponential term appearing in (44) is clearly positive for all
c> 0, on dividing(44) by the exponent and rearranging terms shows that
W0( ) + eW0 ( )1 >0. (45)Therefore, h(c) > 0 and h(c) is an increasing, positive function since h(c) > h(0)= 0
for all c > 0. As the numerator of (40) is also positive, one has du/dc < 0. u istherefore a decreasing function for all c > 0 and since the inverse sine function decreases
monotonically as its argument decreases this implies that max is also a decreasing function
ofc as claimed and confirms, at least for the case of linear resistance, what many expect to
occur to the optimal angle with increasing resistance. Lastly, as W0( (c)) >1, one hasexp[W0( (c)) + 1] 1 > 0, somax is positive for all c > 0, while its decreasing, limitingbehaviour provides the following bound:
0< max 0, (47)
and corresponds to horizontal distances xsuch that
xe1, ddxW0(x) > 0 for x>e1 and consequentlyW0(x)is strictly increasing forx>
e1.
A series expansion for the principal branch can be found using the Lagrange inversion
theorem [42]. The result is
W0(x)=
n=1
(n)n1n!
xn, (A.3)
which converges provided |x| < 1/e.As is seen in the main text, the Lambert W function makes its appearance in the problem
of a projectile projected in a linearly resisting medium through the solution to a particular
class of transcendental equations. Many equations involving exponentials or logarithms which
were formerly considered transcendental can be solved explicitly in terms of the Lambert W
function. In particular, in the case for a linearly resisted projectile its time of flight, range and
optimal angle can all be solved in a closed form in terms of the Lambert W function.
Many other important properties for the function are known. For a brief historical review,properties of the function, its definition when its argument is complex, together with an
overview of some of the areas where the function initially arose, see[43], while more recent
reviews of the function can be found in [44,45]. The function is also to be found listed in
[46]. Numerical values are readily available as it is included as an in-built library function in
3 The function is named in honour of Johann Heinrich Lambert (172877), who in 1758 became the first to considera problem requiring W(x)for its solution. The function, however, would not be formally named in the literature for afurther 230 years.
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Linearly resisted projectile trajectories 165
many modern computer algebra systems. For example LambertW[x] in Maple, lambertw[x]
in Octave andProductLog[x] in Mathematica.
Appendix B. An inequality
In this appendix a proof of the inequality used in section 3.3is presented.
Theorem. Let a function f be continuous on [a, b] and differentiable on (a, b) such that
f(a)= f(b). If f(c)=0 at only a single interior point c (a, b) then f has either a localminimum at c such that f(x) < f(a)= f(b)for all x(a, b)or a local maximum at c suchthat f(x) > f(a)= f(b)for all x(a, b).
Proof. Since f is continuous on [a, b] and differentiable on (a, b) such that f(a)= f(b),Rolles theorem ensures there exists at least one c(a, b)such that f(c)=0.
Now if f(c)=0 at a single interior point c only, then (i) either f(x) > 0 or f(x) < 0for all x (a, c), and (ii) either f(x) < 0 or f(x) > 0 for all x (c, b). Considering eachcase separately.
(i) If f(x) >0 for allx(a, c), f is increasing on the interval(a, c). It follows f(c) > f(a)and fmust decrease on the interval(c, b)if f(b)= f(a)as f is continuous. Accordingly,f(x) < 0 for all x (c, b)and we see that fmust be a local maximum at c. As f onlyhas a single local maximum on the interval(a, b)it follows that f(x) > f(a)= f(b)forallx(a, b).
(ii) Iff(x) 0 for all x (c, b) and we see that fmust be a local minimum at c. As f onlyhas a single local minimum on the interval (a, b)it follows that f(x) < f(a)= f(b)forallx(a, b). This completes the proof.
The result used in the text immediately follows from the above theorem as a corollary.
Corollary. Let a function f be continuous on [a, b] and differentiable on (a, b) such that
f(a)= f(b)=0. If f(c)=0 at only a single interior point c(a, b)then either f(x) >0if f is a local maximum, or f(x)
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