# Geometry and dynamics of motion in ancient Greek science

##### Geometry and dynamics of motion in ancient Greek science

Enrico Canuto, Former Faculty, Politecnico di Torino,  February 28, 2021

What I am writing is my personal opinion that may be adjusted/improved by scholars who studied ancient science. I partly follow the paper of C. Rovelli, Aristotle’s physics, a physicist’s look [1]. I was always impressed by invention of dynamics by Galileo and Newton. Why ancient Greeks did fail in this respect?

###### Tow currents of thought

Two currents of thought about body motion were present in the ancient Greek science.

1. The  geometry of motion aiming to describe the motion of abstract points [1].
2. The Aristotle’s dynamics aiming to describe the motion of real bodies [2].
###### Geometry of motion

The first known treatise on the geometry of motion is the book On the moving sphere  (Περὶ κινουμένης σφαίρας) [3] by Autolykus of Pitane (Αὐτόλυκος ὁ Πιταναῖος; c. 360 – c. 290 BC). The book starts with a definition of the uniform linear motion (Latin translation is akin to original Greek text).

Definitiones
1. Aequabiliter puncta ferri dicuntur, quaecunque aequali tempore aequales ac similes linearum magnitudines percurrunt.
2. Sin autem punctum, quod in linea quadam fertur, aequabiliter duo eius lineae segmenta percurrerit, tempus ad tempus, quo punctum utrumque segmentum percurrit, in eadem ratione erit ac segmentum ad segmentum.

Definitions.
1. Points motion is said to be uniform, when all the equal segments of a line are traveled in the same time.
2. If some point is displaced with a uniform rate along a certain line, and if upon this line we take two segments,they will have the same ratio between them as the times during which the point has traversed these segments (from [1]).

Based on these definitions, he proved several propositions about the motion of a sphere. The first proposition defines the uniform rotation of a sphere about  its  axis. The third proposition explicitly accounts of time.

Propositiones
I. Si sphaera aequabiliter circa axem suum convertetur, omnia in superficie sphaerae puncta, nisi quae in ipso axe sunt, parallelos circulos describent, qui eosdem polos ac sphaera habebunt et perpendiculares ad axem erunt.
III. Si sphaera aequabiliter circa axem suum vertitur, arcus circulorum parallelorum, quos aequali tempore puncta quaedam per eos circulos percurrunt, inter se similes sunt.

Propositions.
I. If a sphere rotates uniformly about its axis, all the points on the surface of the sphere which are not on the axis will trace parallel circles that have the same poles as the sphere, and that are perpendicular to the axis.
III. If a sphere rotates uniformly about its axis, all the points on the surface of the sphere describe in equal times similar arcs on the parallel circles on which they are moving (from [1]).



About one century since the Autolycus’s book, Archimedes of Syracuse (c. 287 – c. 212 BC) employed the composition of the uniform rotation of a line around the extremity which remains fixed and the uniform motion of a point along the line to build up the Archimedean spiral. From On spirals (Περὶ ἑλίκων, [5]):

Figure 1 - Archimedean spiral. Definitions.
I. If a straight line drawn in a plane revolves at a uniform rate about one extremity which remains fixed and returns to the position from which it started, and if, at the same time as the line revolves, a point moves at a uniform rate along the straight line beginning from the extremity which remains fixed,the point will describe a spiral in the plane.

In the geocentric Solar system of Hipparchus of Nicaea (c. 190-120 BC) and of C. Ptolemaeus (c. 100-170), the orbit of planets and Sun around the Earth was predicted by the composition of two uniform clockwise rotations in the ecliptic plane  around a center (eccentric) removed from the Earth center. Each body uniformly rotated inside a small circle called epicycle, and the epicycle center rotated along a large circle (deferent) centered in the eccentric.The resulting trajectory was a deformed circle known as epitrochoid. The model accurately accounted for the apparent retrograde motion of the known superior planets (Mars, Jupiter, Saturn) and inferior planets (Venus, Mercury) as seen from the Earth in the night sky.

The above definitions and propositions were purely geometric (no mathematics is involved) and employed measurable entities like length of segments and time intervals. No real body was involved like in modern kinematics. The velocity of the moving point, as a geometric object,  had no place in their propositions, thus preventing them to near the concept of velocity conservation both in uniform motion, either linear or angular. Motion was only considered as uniform (stationary), thus ignoring the change of motion (transients), which was explained in Aristotle’s dynamics by the sudden action of a mover and the conservation of natural place.

OPINION. In my opinion, lack of the point velocity concept and definition had profound consequence in the science for almost two thousand years, as it reflected on the Aristotle’s dynamics. Before understanding acceleration, one should understand and define velocity! We cannot speak of kinematics but only of geometry of motion, at the best, being mostly confined to astronomy, with the main goal of explaining planetary motion. Further to say, Archimedean composition of motions did not favored any evolution of the distinction between violent and natural motion of the Aristotle’s dynamics for about two thousand years! His written works were fairly unknown in the antiquity and Middle Age. The only significant evolution of the Aristotle’s dynamics was the theory of impetus, first proposed in the VI Century by John Philoponus (Ἰωάννης ὁ Φιλόπονος, c. 490-570), then refined by Arabs commentators of Aristotle and finally by the French philosopher Jean Buridan (c. 1301-1359), before Galileo’s advent.

###### Aristotle’s dynamics

The second current of thought  is the Aristotle’s dynamics (Physics,[4]). The Aristotle’s meaning of motion is rather broad as includes any form of change in nature. We restrict to locomotion, that is the movement from one place to another.

The fundamental concept is the conservation of the natural place. Each of the five substances, Earth, Air, Water, Fire and Ether, has its natural place, the Earth ground, the sphere of Water surrounding the Earth, the Atmosphere (Air), the Fire sphere above the atmosphere and the Heavens (Ether). As soon as a body made by a substance is displaced from its natural place, it tends to return to its place along a vertical trajectory (natural motion). Not the Heavens which rotate of uniform motion around the center of the Universe (the Earth).

Beside natural motion and distinct, Aristotle defined the violent or unnatural motion caused by a mover. A body which undergoes a violent motion, must be continuously forced to move by a mover. If the motive force acted by the mover ceases, for instance  because the contact between mover and body (think of an arrow propelled by a bow) ceases, the violent motion ceases too.

Physics, 254b12-254b32. Of things which move in their own right, some derive their motion from themselves, others from something else: and in some cases their motion is natural, in
others violent and unnatural.
...  And the motion of things that derive their motion from something else is in some cases natural, in others unnatural: e.g. upward motion of earthy things and downward
motion of fire are unnatural. 

Based on this definition, Aristotle could bypass the principle of the  conservation of motion at the price (for us, not for Aristotle who surrounded the Earth by air and water spheres) that air and water (the medium) became sequential movers triggered by the motive force of the prime mover, thus allowing a body (think of an arrow, a projectile in general) to move once separated by the prime mover (the bow, the propeller).  Since vacuum does not contain any substance like air and water, it could not allow locomotion and therefore Aristotle denied its existence, or better assumed it as unnecessary. The theory, known as antiperistasis,  has up to now entrained uncountable disputes and commentaries.

OPINION. In my opinion, the need of a continuous (actually discrete) mover provided by the medium (air and water) was necessary because of the axioms:

1. the conservation of the natural place asks for a natural motion, distinct from the violent motion,
2. composition between natural and violent motions cannot occur,
3. violent motion changes and comes to ceasing, because the motive force of mover(s) decreases,
4. when violent motion ceases, natural motion may take place (think of a projectile),
5. nothing is said about the initial change of motion from rest or from violent to natural, implying ignorance of the conservation of motion and of the body velocity (see conclusions on the geometry of motion),
6. as a partial surrogate of the conservation of motion (think of a projectile), air and water (the medium), triggered by a prime mover, may act as movers, but their motive force tends to progressively cease.

The above axioms are well illustrated by the trajectory of a mortar projectile in the drawing of the Dutch mathematician D. Santbech (active in the XVI century), as it appeared in his book Problematum astronomicorum et geometricorum sectiones septem [6].

Figure 2 - Aristotelian projectile trajectory (from Wikipedia).
The straight line up to the point K is the Aristotelian violent motion, which does not cease because of the gravity, but because either the fluid motive force (Aristotle, the 'flame' around the violent trajectory) or the impetus (Philoponus, see below) progressively diminishes, until the vertical natural motion takes place. Neither concept or observation of (local) conservation of motion (velocity) emerges.

Some progress in the drawing of the projectile trajectory was done by the Venitian mathematician N. Fontana (1499-1557, known by the nickname Tartaglia) in the Nova Scientia (1532), likely to agree with observations, by connecting violent and natural trajectories with a circular arc, still part of the violent motion, not to contradict the Aristotelian distinction between violent and natural motion as stated in the Fifth Proposition.

No equally heavy body can travel for an interval of time or a space with a motion mixed of violent and natural motion.

So immense and constraining was Aristotle’s authority! Composition of motions was still to come! An ‘equally heavy body’, as defined by Tartaglia,  is an ideal body whose motion does not suffer air resistance.

Figure 3 - Drawing of a projectile trajectory in Nova Scientia (from [7])
The circular arc as part part of the violent motion (from A to D) was justified as follows.
"Assuming (as the opponent says) the [body] could travel some part with violent and natural motions mixed together, which may be part CD, it follows therefore that the mentioned body, while going from point C to point D, increases its velocity according to the ratio by means of which it shares a natural motion (because of the first proposition). Likewise, it decreases its velocity according to the ratio by means of which it shares a violent motion (because of the third proposition). It is absurd that the mentioned body increases and decreases its velocity at the same time"(from [7]).

In essence , he denies the composition of motions. But also the proponents, from John Philoponus to Jean Buridan, of the theory of impetus as opposed to Aristotle’s antiperistasis did not make much better in solving the projectile motion. Not to diminish their concepts and observations, they just made internal to body the continuous motive force acted by the medium itself, in agreement with the medium resistance to body motion and not vice versa.

###### At the end … Galileo!

One of the most significant experiment and geometrical exercise of G. Galilei (1564-1642), was the composition of the linear horizontal uniform motion (conservation of motion) with the vertical naturally accelerated motion (the natural motion), which provided him, first among all, with the parabolic trajectory of the projectile motion. The problem was mentioned among others in the unpublished essay De motu antiquiora  (c. 1592). At the time of writing the Dialogue Concerning the Two Chief World Systems (Dialogo sopra i due massimi sistemi del mondo, 1632) he recognized the great significance of problem and solution.

Dialogo sopra i due massimi sistemi del mondo, Part IV.
THE MOTION OF PROJECTILES
In the preceding pages we have discussed the properties of uniform motion and of motion naturally accelerated along planes of all inclinations. I now propose to set forth those properties which belong to a body whose motion is compounded of two other motions, namely, one uniform and one naturally accelerated; these properties, well worth knowing, I propose to demonstrate in a rigid manner.  This is the kind of motion seen in a moving projectile. ...
THEOREM 1, PROPOSITION I
A projectile which is carried by a uniform horizontal motion compounded with a naturally accelerated vertical motion describes a path which is a semi-parabola. 
###### Epilogue

Then modern science started mixing mathematics (Greek science) with experiments (fairly unknown to Greek science). Experiments asked for accurate measurements and environment control. The keys to modern technoscience: measurement and control, from nature to mathematics, from math to nature. In the mid, to close the loop, the Embedded model control [8].

Hence, as a first conclusion, we conjecture that the absence of adequate instrumentation and experimentation, could have hidden geometry and timing of the rather common projectile motion, notwithstanding a rather developed mathematics, leading, in the end,  to contrived  speculations.  Time was measured by portable water clocks, clepsydras (‘water thief’), with resolution better than minute. Hellenistic physician Herophilos (335-280 BC) employed a portable clepsydra to measure pulse rate [9]. Further, geometry of conic sections was known since the same epoch of Aristotle, by the work of Menaechmus (380-320 BC). Only controlled experiments might favor a wise composition of mathematics and measurements.

###### References

[1] M. Bacelar Valente, Geometry of motion: some elements of its historical development, ArtefaCToS. Revista de estudios de la ciencia y la tecnología, Vol. 8, No. 2, 2019, pp. 5-26, DOI: http://dx.doi.org/10.14201/art201982526.

[2] C. Rovelli, Aristotle’s physics: a physicist’s look, Journal of the American Philosophical Association, Vol. 1, 2015, pp. 23-40, DOI: 10.1017/apa.2014.11.

[3] Autolycus, On The Moving Sphere, from the Million Books Project (Greek with Latin translation) .

[4] Aristotle, Physics, Public Domain English Translation  by R. P. Hardie and R. K. Gaye, http://people.bu.edu/wwildman/WeirdWildWeb/courses/wphil/readings/…

[5] R. Netz, The works of Archimedes: Volume 2. On spirals. Translation and commentary, Cambridge University Press, 2017.

[6] S.M. Walley, Aristotle, projectiles and guns, April 2018, from https://www.researchgate.net/publication/324182018_Aristotle_projectiles_and_guns

[7] M. Valleriani,  Metallurgy, Ballistics and Epistemic Instruments: The Nova Scientia of Nicolò Tartaglia, 2013, pp. 169-181.

[8] E. Canuto, C. Novara, L. Massotti, D. Carlucci, C. Perez Montenegro, Spacecraft dynamics and control: the embedded model control approach, Butterworth-Heinemann, 2018.

[9]  von Staden H. (ed. trans.) Herophilos: The Art of Medicine in Early Alexandria. Cambridge University Press, 1989.

##### Appendix
###### Aristotle’s difficulties

To better grasp difficulty and necessity encountered by Aristotle because of the above assumptions and neglects, let us include the  paragraph about the projectile motion from Physics, Book VIII, Chapter 10 [4].

If everything that is in motion with the exception of things that move themselves is moved by something else, how is it that some things, e.g. things thrown, continue to be in motion when their mover is no longer in contact with them? If we say that the mover in such cases moves something else at the same time,that the thrower e.g. also moves the air, and that this in being moved is also a mover, then it would be no more possible for this second thing than for the original thing to be in motion when the original mover is not in contact with it or moving it: all the things moved would have to be in motion simultaneously and also to have ceased simultaneously to be in motion when the original mover ceases to move them. Therefore, while we must accept this explanation to the extent of saying that the original mover gives the power of being a mover either to air or to water or to something else of the kind, naturally adapted for imparting and
undergoing motion, we must say further that this thing does not cease simultaneously to impart motion and to undergo motion: it ceases to be in motion at the moment when its mover ceases to move it, but it still remains a mover, and so it causes something else consecutive with it to be in motion, and of this again the same may be said. The motion begins to cease when the motive force produced in one member of the consecutive series is at each stage less than that possessed by the preceding member, and it finally ceases when one member no longer causes the next member to be a mover but only causes it to be in motion. The motion of these last two - of the one as mover and of the other as moved - must cease simultaneously, and with this the whole motion ceases. Now the things in which this motion is produced are things that admit of being sometimes in motion and sometimes at rest, and the motion is not continuous but only appears so: for it is motion of things that are either successive or in contact, there being not one mover but a number of movers consecutive with one another: and so motion of this kind takes place in air and water. 
###### Geometry of projectile’s motion

Let us recall the geometry of the projectile motion, in absence of  air resistance, by denoting the horizontal and vertical coordinates by $x$ and  $z$. The vertical profile as a function of $x$ holds:

$z(x)=4&space;H\frac{x}{L}\left&space;(1-\frac{x}{L}&space;\right&space;),&space;0\leq&space;x\leq&space;L$

where $H$ is the peak height and $L$ the traveled horizontal distance. By adding the travel time $T$, we obtain the three motion parameters: the launch elevation (from horizontal plane)  $\theta$, the launch velocity magnitude $v_0$ (related to propulsion) and the gravity acceleration $g$:

$tan\theta=\frac{4H}{L},&space;v_0=\frac{\sqrt{L^2&space;+16&space;H^2}}{T};&space;g=\frac{8H}{T^2}$

Derivation: state equations

$\begin{matrix}\dot{x}(t)=v_x(t),x(0)=0&space;\\&space;\dot{v}_x(t)=0,v_x(0)=v_0&space;cos&space;\theta\\&space;\dot{z}(t)=v_z(t),z(0)=0\\&space;\dot{v}_z(t)=-g,v_z(0)=v_0&space;sin&space;\theta&space;\end{matrix}$Parabolic solution

$\begin{matrix}x(t)=tv_0(t)cos&space;\theta&space;\\&space;z(t)=tv_0(t)sin\theta-\frac{g}{2}t^2\\&space;z(x)=x&space;tan\theta-\frac{g}{2&space;v_0^2&space;cos^2&space;\theta}x^2&space;\end{matrix}$

Geometric parameters and timing

$\begin{matrix&space;}T=t(z=0)=\frac{2&space;v_0&space;sin\theta}{g}&space;\\&space;L=x(T)=\frac{v_0^2&space;2&space;cos&space;\theta&space;sin\theta}{g}&space;\\&space;H=z(T/2)=\frac{v_0^2&space;sin^2\theta}{2g}\\&space;v_0^2&space;cos^2&space;\theta=\frac{L^2}{T^2},&space;g=&space;\frac{8H}{T^2}\end{matrix}$

###### Aristotle and transients

Let us recall the following Aristotle’s sentence, from [10]:

“Bodies of different weight […] move in one and the same medium with different speeds which stand to one another in the same ratio as the weights so that, for example, a body which is ten times as heavy as another will move ten times as rapidly as the other”.

The relevant state equation (Newton’s) of the falling speed (positive toward nadir) under gravity and air resistance and buoyancy holds

$\begin{matrix&space;}&space;\dot{v}(t)=&space;g&space;(1-\frac{\rho_{fluid}}{\rho_{body}})-\frac{c_D}{2}\frac{A&space;\rho_{fluid}&space;}{m}v^2(t),&space;v(t_0)=v_0&space;\textup{[m/s]}\\&space;v_\infty&space;\simeq&space;\sqrt{2\frac{mg}{c_D&space;A}&space;(\frac{\rho_{body}}{\rho_{fluid}}-1)}&space;\end{matrix}$

where the $c_D&space;\simeq&space;2$  is the drag coefficient and $\beta=m(c_D&space;A)^{-1}$ is the ballistic coefficient. The steady state velocity $v_{\infty}$ shows to be actually proportional to  the square root of the weight $\sqrt{mg}$  and not simply to the weight, and through the shape coefficient $c_dA$  and the density ratio $\frac{\rho_{body}}{\rho_{fluid}}$.

The equation solution is the following

$\begin{matrix&space;}\frac{v(t)}{v_{\infty}}=&space;\textup{tanh&space;}(\frac{\rho_{fluid}&space;}{\beta}\frac{v_{\infty}}{2}&space;t&space;)\\&space;\lim_{t&space;\rightarrow&space;\infty}\frac{v(t)}{v_{\infty}}=1&space;\\&space;\lim_{t&space;\rightarrow&space;0}\frac{v(t)}{v_{\infty}}=\frac{t}{\tau}&space;,&space;\tau=\frac{2&space;\beta}&space;{\rho_{fluid}v_{\infty}}=\frac{\sqrt{2&space;\beta}}&space;{\rho_{fluid}}&space;\frac{1}{\sqrt{&space;g(\frac{\rho_{body}}{\rho_{fluid}}-1)}}\end{matrix}$

where the steady state velocity is reached after a transient defined by the time constant $\tau$. In the air  $\rho_{air}&space;\simeq&space;1&space;\:&space;\textup{kg/m}^3$ and $\rho_{body}&space;\gg&space;\rho_{air}$, which allows the simplification

$\tau&space;\simeq&space;\sqrt{\frac{2&space;\beta}{g&space;\rho_{body}}}=\sqrt{\frac{2&space;V}{g&space;C_D&space;A}}\rightarrow&space;\tau_{sphere}&space;\simeq&space;\sqrt{\frac{2&space;r}{3g&space;}}\ll&space;1&space;\:\textup{&space;s},&space;2r&space;<&space;1&space;\textup{m}$

The original equation can be rewritten in terms of $\tau$ and of the dimensionless ratio$w=\frac{v}{v_\infty}$

$\begin{matrix&space;}&space;\tau&space;\dot{w}(t)\simeq&space;\tau&space;\frac&space;{g&space;}{v_\infty}-w^2(t),&space;w(t_0)=w_0&space;\textup{[m/s]}&space;\end{matrix}$

The above decomposition into transient and steady state, might suggest that Aristotle had no interest (he ignored) in the response transient, since of course he knew that initial speed was zero (better, that the body was at rest), but ignored (or misled) concept and meaning of speed difference, although he stated that motion is the result of a cause or of the necessity of returning to its natural place. He only knew concept and measurement of weight (by comparison through balance), also measurement of speed was given indirectly as the distance traveled in time from motion start to motion halt, thus ignoring speed variations (another form of transient).

In modern terms, transient may be neglected when time constant (here $\tau$ [s]) happens to be much smaller than observation time or time resolution, which amounts to assume $\tau&space;\dot&space;w(t)\ll&space;\frac{\tau&space;g}{v_\infty}-w(t)^2$  . Time was measured by portable water clocks, clepsydras (‘water thief’), with resolution round minute, likely, implying ignorance of shorter transients. .

We may guess that ignorance of transients prevented them to clearly define the cause of the motion as a quantity independent of speed. In fact, by neglecting transient, the cause of motion can be treated as proportional to the speed itself, a quantity known as the impulse of force. Formally, the impulse of force is the integral of a time-limited force profile (let us warn that time duration may be unlimited as in Gaussian probability density, but integrable) like that of spacecraft thrusters, with unit [Ns] [8]. When the impulse duration is short, it can be formulated through a Dirac delta $I_0&space;\delta(t)$, where $I_0$  is the impulse (Ns) and $\delta(t)$ is a temporal function with unitary integral whose time duration is brought to be negligible while conserving the integral. The unit of $I_0&space;\delta(t)$ is [N], the unit of $\delta(t)$ is [1/s] and the unit of the integral equal to $I_0$is [Ns] (impulse of force). $\delta(t)$, being unlimited in amplitude, makes negligible all the other terms in a state equation as follows:

$\dot&space;v(t)=\frac{I_0}{m}&space;\delta(t)&space;+&space;\textup{negligible&space;terms&space;},&space;v(t_0-h)=v_0$

where $h$ is the negligible half duration of $\delta(t)$

Integration provides

$v(t_0+h)=v(t_0-h)&space;+I_0/&space;m,&space;\textup{[m/s]}&space;\rightarrow&space;\Delta&space;v=v(t_0+h)-v(t_0-h)=I_0/&space;m$

The velocity difference is known as DeltaV  and is normally employed in spacecraft dynamics [3] .

As a result, acceleration $\dot&space;v(t)$ and the force as the cause disappear in favor of the force impulse (DeltaV=speed difference). Newton’s equation may be replaced by the kinematic equation of the locomotion

$\begin{matrix}&space;\dot&space;s(t)=\Delta&space;v(t)=I_0(t)/m,&space;s(t_0)=s_0&space;\\&space;s(t)=s_0+m^{-1}\int_{t_0}^{t}I_0(\tau)d\tau&space;\end{matrix}$

The above discussion may also explain the concept of the motion as potentially existing in falling bodies. You just need a cause like a trigger to start bodies to fall: then motion naturally continues (free response to impulse of force) because of a body inherent quality. As a further example, the concept applies to arrow motion (different from fall), likely, the fastest motion they could observe.

###### Derivation of the body falling equation

Let us start from

$\begin{matrix&space;}&space;\dot{v}(t)=&space;b^2(a^2&space;-v^2)&space;g&space;(1-\frac{\rho_{fluid}}{\rho_{body}})-\frac{c_D}{2}\frac{A&space;\rho_{fluid}&space;}{m}v^2(t),&space;v(t_0)=v_0&space;\textup{[m/s]}\\&space;v_\infty=&space;a=&space;\sqrt{2\frac{mg}{c_D&space;A}&space;(\frac{\rho_{body}}{\rho_{fluid}}-1)},&space;b=\sqrt{\frac{c_D&space;A&space;\rho_{fluid}}{2m}&space;}&space;\end{matrix}$

The integration provides

$\begin{matrix&space;}\frac{1}{a}\textup{atanh}(\frac{v}{a})=\frac{1}{a}\textup{atanh}(\frac{v_0}{a})+b^2(t-t_0)&space;\\&space;\frac{v(t)}{v_{\infty}}=&space;\textup{tanh&space;}(\textup{atanh}(\frac{v_0}{v_{\infty}})+v_{\infty}b^2(t-t_0)&space;)\end{matrix}$

By assuming $v_0=0,&space;t_0=0$, the solution simplifies to

$\begin{matrix&space;}\frac{v(t)}{v_{\infty}}=&space;\textup{tanh&space;}(\frac{\rho_{fluid}&space;}{\beta}\frac{v_{\infty}}{2}&space;t&space;)\\&space;\lim_{t&space;\rightarrow&space;\infty}\frac{v(t)}{v_{\infty}}=1&space;\\&space;\lim_{t&space;\rightarrow&space;0}\frac{v(t)}{v_{\infty}}=\frac{t}{\tau}&space;,&space;\tau=\frac{2&space;\beta}&space;{\rho_{fluid}v_{\infty}}\end{matrix}$