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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:[email protected]

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:[email protected]://stacks.iop.org/EJP/33/149http://stacks.iop.org/EJP/33/149mailto:[email protected]://dx.doi.org/10.1088/0143-0807/33/1/013


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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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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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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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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


12/19


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.


18/19


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)


19/19

166 S M Stewart

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