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Matter Gravitates, but DoesGravity Matter?C. W. GroetschPublished online: 13 Feb 2011.

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PRIMUS, 21(2): 142–148, 2011Copyright © Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970903528094

Matter Gravitates, but Does Gravity Matter?

C. W. Groetsch

Abstract: The interplay of physical intuition, computational evidence, and mathe-matical rigor in a simple trajectory model is explored. A thought experiment based onthe model is used to elicit student conjectures on the influence of a physical parameter;a mathematical model suggests a computational investigation of the conjectures, andrigorous analysis at the level of elementary calculus is used to settle the issue.

Keywords: Torricelli’s law, air resistance, implicit differentiation, intermediate valuetheorem, Rolle’s theorem, quadratic functions.

1. INTRODUCTION: A NAÏVE MODEL WITH A SURPRISE

Suppose a small hole is opened at a height h above the base of a tank thatis filled with water to a depth H > h. Water is impelled through the holewith a velocity that depends on the strength of gravity, g, and the size ofthe hydraulic head, H − h. The horizontal velocity of the leading droplet ofwater that emerges from the hole is given by Torricelli’s law: v = √

2g(H − h).This physical law amounts to a balancing of the energy budget. When the firstinfinitesimal droplet of mass m emerges, work in the amount of mg(H − h)is required to restore the tank to its initial state. This work (= change inpotential energy) is converted into the kinetic energy 1

2 mv2 of the emergentdroplet. Equating the loss of potential energy to the gain of kinetic energy givesTorricelli’s law.

Ignoring air resistance, the horizontal distance covered by the drop intime t is

x = vt = √2g(H − h) × t,

while, by Galileo’s Law of Fall, its altitude is

Address correspondence to C. W. Groetsch, School of Science and Mathematics, TheCitadel, Charleston, SC 29409, USA. E-mail: groetschc1@citadel.edu

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Matter Gravitates 143

y = h − g

2t2.

The time of flight is then T = √2h/g, and so the horizontal range of the spurt

is,

R = vT = 2√

h(H − h).

At this point, I ask students if there is anything about this result that sur-prises them. The response almost invariably is something like: “No, we did themath, and that’s what we got.” I then ask what physical mechanism makes thewhole thing “go.” Usually I hear that gravity ‘pulls’ the drop through the hole,sending it on its way. Now they get it: gravity is the only motivator, yet therange is independent of gravity strength — it’s the same on Jupiter as it is onEarth!

A little reflection explains this. Increasing gravity strength has two effects:it increases the horizontal velocity, but it also increases the descent velocityand hence decreases the flight time. These two effects exactly cancel eachother, mathematically and physically. But students are quick to point out thatthe model is quite naïve since it ignores air resistance. What if there is airresistance? Will gravity then assert itself in the range problem and, if so, howwill the range depend on g? Unscientific surveys of my students on this pointusually result in about half expressing the belief that gravity continues to haveno effect; the remaining students split roughly equally between those who feelthat increasing gravity strength will increase the range and those who are per-suaded that it will decrease the range. In any event the urge to refine the modelto include air resistance and test the conjectures mathematically is irresistible.

2. A SIMPLE REFINED MODEL

The simplest representation of a resistive medium, which has been found tobe quite suitable for low velocities [1], is the first-order model in which theresistive force is directly proportional, but oppositely directed, to the velocity.Since we are interested only in the dependence on g, there is no harm in savingsome ink by taking the hydraulic head H − h to be 1. The horizontal veloc-ity of the emergent drop is now a simple function of g: v = v(g) = √

2g, andconveniently, v′(g) = 1/v. We may also assume that the emergent drop has unitmass. Taking the resistive constant of proportionality to be k, Newton’s Law ofMotion then provides our refined model:

x = −kx, x(0) = v, x(0) = 0y = −g − ky, y(0) = 0, y(0) = h.

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144 Groetsch

We honor Newton here by using his notation, which is standard in the dynam-ics community: the dots signify derivatives with respect to time. These lineardifferential equations may be solved by elementary techniques (see, e.g., [2]),yielding

x = v

k(1 − e−kt)

y = h − g

kt + g

k2(1 − e−kt). (1)

Eliminating the time parameter gives,

y = h + g

k2ln

(1 − kx

v

)+ g

kvx,

and on setting y = 0 we find that the range, R, satisfies

0 = kR

v+ hk2

g+ ln

(1 − kR

v

). (2)

I encourage students to use this equation to test their conjectures on thedependence (or independence) of the range R on the strength of gravity g. Theycan do this by plotting, for given fixed positive values for k and h, and variousvalues for g, the expression on the right hand side of (2) (remembering thatv = √

2g) and noting the behavior of the root R. Or, they can solve (2) bya numerical method for various positive values of g. In any case the computa-tional evidence suggests that gravity does matter: the range certainly appears toincrease as the strength of gravity increases. Some students accept this evidenceas “proof” that the range increases with gravity strength and are content toleave it at that. The more discriminating student appreciates that (2) is derivedfrom a mathematical model and that a mathematical proof based on this modelwould be a more satisfying outcome. This presents the opportunity of mar-shaling several fundamental concepts from elementary analysis to construct aproof. Among these are the intermediate value theorem, implicit differentia-tion, and some basic properties of quadratic functions. Our goal in this noteis purely mathematical. We aim to provide a rigorous elementary proof that,in the model described by (1), the range is an increasing function of gravitystrength g.

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Matter Gravitates 145

3. SO HOW DO WE PROVE IT?

The range of the leading drop is given by R = x(Tg), where Tg is the flight time,that is, the positive time at which the y-coordinate vanishes. In other words, theflight time is the positive solution of the equation

Tg = kh

g+ 1

k

(1 − e−kTg

).

That this equation has a unique positive solution is clear from a graph; thegraph also points the way to an easy analytical proof of this fact.

Students know that to prove R(g) = x(Tg) increases with increasing g it isenough to show that dR

dg > 0. The problem is that R is given only implicitly by(2). But this suggests that students might use skills of implicit differentiationhoned in their calculus classes. Before proceeding, it is helpful to notice someobvious groupings in (2) that suggest the introduction of the variable s = kR/v.Equation (2) then calls for a root of the function

f (s) = s + a/

g + ln (1 − s), (3)

where a = hk2 is a positive constant. Note that by (1),

0 < s = kx(Tg)/v = 1 − e−kTg < 1,

and so we are led to seek a root of (3) in the interval (0, 1). For any givenpositive g, the intermediate value theorem guarantees the existence of a rootsince f (0) > 0 and lims→1− f (s) = −∞. Also, since f ′(s) < 0 for 0 < s < 1,Rolle’s Theorem ensures that the root is unique. If we denote by r(g) the uniqueroot of (3) in (0, 1) corresponding to a given positive number g, then analysisof the range R(g) may be transferred to the function r(g), where

r(g) = kR(g)

v.

A key point in our discussion is the observation that, for a given s ∈ (0, 1):

r(g) > s if and only if f (s) > 0, (4)

since f is strictly decreasing on the interval (0, 1) and r(g) is the unique root off in (0, 1).

An expression for r′(g) may be had courtesy of implicit differentiationof (3):

0 = d

dgf (r(g)) = r′(g) − a

/g

2 − r′(g)/(1 − r(g)),

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146 Groetsch

and hence

r′(g) = − a

g2

1 − r(g)

r(g). (5)

(Students who have qualms about the differentiability of r may be referred tothe implicit function theorem (e.g., [3]) for reassurance). Since v′ = 1/v andkR(g) = r(g)v(g), we have,

kR′(g) = r′(g)v + r(g)/v,

and therefore R′(g) > 0 if and only if

0 < r′(g)v2 + r(g) = 2gr′(g) + r(g).

Substituting the result for r′(g) given in (5) into this inequality shows thatR′(g) > 0 if and only if

0 < −2a

g

1 − r(g)

r(g)+ r(g).

On multiplying by r(g) and rearranging a bit, we find that R′(g) > 0 if andonly if

r(g)2 + 2a

gr(g) − 2a

g> 0.

The quadratic expression

x2 + 2a

gx − 2a

g(6)

is concave-up and takes a negative value at x = 0, therefore the number r(g)∈ (0, 1) satisfies this quadratic inequality if and only if its is greater than thepositive root, that is, ρ < r(g) < 1, where ρ is the positive root of the quadraticexpression (6). The quadratic formula gives us this positive root:

ρ = −a

g+

√(a

g

)2

+ 2a

g.

Note that ρ < 1 since (a/g)2 + 2a/g < (1 + a/g)2. This expression for ρ gives

(ρ + a

g

)2

=(

a

g

)2

+ 2a

g,

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Matter Gravitates 147

or equivalently

ρ2 + 2a

gρ = 2a

g,

and hence

2a

g= ρ2

1 − ρ. (7)

But, as was observed in (4), the condition r(g) > ρ is equivalent to f (ρ) > 0.However, using (7), we find that

f (ρ) = ρ + a

g+ ln (1 − ρ)

= ρ + ρ2

2(1 − ρ)+ ln (1 − ρ),

which is positive for any ρ ∈ (0, 1). So the case is closed, at least for thissimple resistance model: dR

dg > 0, the range is an increasing function of gravitystrength.

4. CODA

I have presented material from this note in a variety of courses. The naïvemodel of the first section has been used in elementary calculus courses with afollow-up exercise on determining the hole height h that produces the max-imum range. Aspects of the material in the second section provide usefulillustrations of techniques from elementary differential equations courses, aswell as a case study for courses in mathematical modeling and numerical anal-ysis. A junior-level course in mathematical analysis is an ideal venue for thetreatment in section three which closes the loop on the modeling, refinement,conjecture, computation, and analysis process, with a rigorous proof that R(g)is an increasing function of g.

REFERENCES

1. Battye, A. 1991. Modelling air resistance in the classroom. TeachingMathematics and Its Applications. 10: 32–34.

2. Boyce, W. E., and R. C. DiPrima. 2005. Elementary Differential Equations,8th Edition. New York: John Wiley.

3. Courant, R., and F. John. 1974. Introduction to Calculus and Analysis, Vol.II. New York: Wiley-Interscience.

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148 Groetsch

BIOGRAPHICAL SKETCH

One wouldn’t guess it from his surname, but C. W. Groetsch was born andraised in the Irish Channel neighborhood of New Orleans. After serving manyyears on the faculty of the University of Cincinnati, he replanted his southernroots in Charleston, South Carolina, where he occupies the Traubert Chair atThe Citadel and serves as dean of the School of Science and Mathematics.Aside from his nine grandchildren, his chief diversion from administrative lifeis the joy he gets from blending history, physical science, and mathematics inhis classroom teaching.

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