# Preface

This is a
tutorial made solely for the purpose of education and it was designed
for students taking Applied Math 0330. It is primarily for students who
have very little experience or have never used
*Mathematica* before and would like to learn more of the basics for this computer algebra system. As a friendly reminder, don't forget to clear variables in use and/or the kernel.

Finally, the commands in this tutorial are all written in bold black font,
while *Mathematica* output is in normal font. This means that you can
copy and paste all commands into *Mathematica*, change the parameters and
run them. You, as the user, are free to use the scripts for your needs to learn the *Mathematica* program, and have
the right to distribute this tutorial and refer to this tutorial as long as
this tutorial is accredited appropriately.

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# Applications

**Vertical Motion**

It is known from elementary physics
that, in the absence of air friction, a ball thrown up from the ground
into earth's atmosphere with initial speed *v*_{0} would attain a
maximum altitude of \( v_0^2 /(2g) .\) In this case the return time is
\( 2\, v_0 /g, \) independent of the ball's mass. Here *g* is the
acceleration due to gravity. If the ball is thrown up from
altitude *y*_{0} (which we later assume to be zero), then the time *T*_{0}
spent traveling is given by

The presence of air influences the ball's motion: it experiences two forces acting on it---the force of gravity and the air resistance force. Let's define the symbol T as follows:

Without air resistance, the object travels farther up than with air
resistance. On the way down, without air resistance the object
travels a larger distance, but it also gathers more speed.
A natural question is, which travel time (with air
resistance vs. no air resistance) is larger? Also, it is of interest
to find the maximum altitude \( y_{\max} \) of the ball, the time
*T*_{max} to reach maximum altitude, and the time
\( T_{\text{down}} \) to return back from *y*_{max}. Therefore, \( T_{\max} +
T_{\text{down}} = T_{\text{total}} \) (the total time the ball spent in
the air). The landing velocity is denoted by \( v_{\ell} . \)

**Example**: model for air resistance.
Air resistance is the force that acts in the direction opposite to the
motion of an object through air.
Air resistance depends on the shape, material, and orientation of the object,
the density of the air, and the object's relative speed.

We would like to think that there is a nice formula for the air resistance in terms of speed and other variables. Such a formula would help in making calculations and predicting various quantities. A starting point for obtaining such a formula is our everyday experience. Based on our experience, a reasonable assumption to make\footnote{Our intuition based on everyday experience is limited to a small range of conditions. This may lead to erroneous assumptions.

It has been observed that,
*under suitable conditions*,
the magnitude of the air resistance is proportional to a power of the
speed s=|v|:

*v*is velocity, and both

*k*and

*p*are positive constants. For very small objects, such as a speck of dust (about 1 micrometer or 0.001mm), p=1 seems to give a reasonable formula for the air resistance. For larger, human scale objects moving at relatively large speed, p=2 works better. Therefore, the magnitude of the air resistance

*F*as a function of velocity

*v*is assumed be given by formula (M).

The air resistance force depends on the velocity (*v*) of the object at
time *t*, so let us denote this force with the symbol *F(v)*. Note
that the air resistance, force *F(v)*, always acts in the direction
opposite to the motion. Therefore, *F(v)* acts in the down (negative)
direction when the ball is moving up, and it acts in the up (positive)
direction when the ball is moving down.
If we measure the displacement *y = y(t)* vertically upwards from the
ground, then \( v= {\text d}y/{\text d}t = \dot{y} \) is the velocity of
the object. Newton's law of motion for the ball *on the way up*
gives the differential equation

*on the way down*,

*on way up*becomes

*on the way down*,

To find an equation for $v_{\ell}$, the landing velocity, we rewrite
\refeq{vert.3}
as \( {\text d}\,t = - {\text d}v/(g +F(v)/m) \)
and integrate both sides from t=0 and *v = v*_{0} to \( t=T_{\max} \) and
*v=0*. Here \( T_{\max} \) is the time to reach the maximum altitude
\( y_{max} , \) which is also the time to have velocity *v=0*.
We obtain,

To find an equation for $v_\ell$, we rewrite \refeq{vert.3}
as \( v\,{\text d}t = -v\, {\text d}v/(g +F(v)/m) \)
and integrate
both sides from t=0 and *v = v*_{0} to \( t=T_{\max} \) and *v=0*.
Here \( T_{\max} \) is the time to reach the maximum altitude
\( y_{\max} , \) which is also the time to have velocity *v=0*.
Using the fact that the integral of the velocity is the displacement,
we obtain,

Solve[Integrate[v/(-g + 0.01*v^2), {v, 0, vl}] == Integrate[v/(g + 0.01*v^2), {v, 50, 0}], vl]

We would like to
determine the ratio:
\[
\gamma = T/T_0 = Tg/ \left( v_0 + \sqrt{v_0^2 +2 x_0 g}\right),
\]
where T time in air with air resistance and *T*_{0} is the time in air
without air resistance.

For a tennis ball thrown upward with the initial velocity *v*_{0} =10,
it is possible to find *x*_{0} that γ > 1 when p=0.9. In
general, it is unknown for what values of p< 1, *x*_{0}, and *v*_{0} we can
achieve γ > 1.

**Example**: We consider a model of falling object, say a tennis ball, to
a flat surface that moves up and down periodically. Using vertical axis
directed upward, we denote *v(t)* as the velocity of the ball and *y(t)*
as its position/height at time *t*.
It has been observed that, *under suitable conditions*,
the magnitude of the air resistance is proportional to the power of speed*s=|v|*:

*F(v)*always acts in the direction opposite to the motion. Therefore,

*F(v)*acts in the down (negative) direction when the ball is moving up, and it acts in the up (positive) direction when the ball is moving down.

Suppose that initially at *t=0*, the ball of mass *m* is dropped from the
altitude \( y=h > 1 \) without initial velocity. At the same time, it is assumed
that the floor starts moving according to the formula \( z= \sin \omega t. \)
When elastic ball hits a hard flat surface, it bounces back with the same
velocity.
It is assumed that the collision is totally elastic, so the ball loses
no kinetic energy in the collision, and its speed after collision is the same
as before the collision. At this point, ignore the time needed for the ball
to be deformed during collision before fully rebounded and has lifted off
from the surface instantly. Hence the ball can be treated as a rigid body with
negligible deformation during impact.

After collision, the ball climbs up until its velocity becomes zero, and then the ball falls vertically downward under the influence of gravity, hits the the moving floor, and bounces back.

## Derivation of a differential equation

Newton's law of motion for the ball *on the way down* is

*on the way up*

*g*is the acceleration due to gravity. Since we assume \( F(v) = k \, |v|^p ,\) the equation of motion

*on the way down*becomes

*on way up*

*p=2*for a tennis ball, the above differential equations can be integrated using separation of variables:

*v(t)*is the derivative of ball's position

*y(t)*, you may need to integrate

## Input parameters

\( g \approx 9.806 \) m/sec^{2} |
the acceleration due to gravity near sea level at 45 deg. latitude; |
---|---|

m=0.08 | mass of the object, in kg because a tennis ball is about 80 grams; |

k=0.02 | drag coefficient, positive; |

p=2 | power of the speed term in the resistance force; |

ω = π | frequency of the oscillating floor; |

\( y_0 =h > 1 \) | initial altitude, positive, in meters. |

## Derivation of solution

A ball that is dropped from height *h*> 1 can be described by its
velocity *v(t)* and position *y(t)*:

*v(t)*on first stage of falling from hight

*h*> 1 explicitly:

*v(t)*:

*Mathematica*confirms:

Integrate[A*Tanh[B*t], t]

k := 0.02; g := 9.806; m := 0.08; h0 := 2; (* h0 is the initial height *)

FindRoot[Sin[\[Pi] t] == h0 - 4*Log[Cosh[1.5657266683556232*t]], {t, 0.4}]

vv[t_] = omega*Cos[omega*t]

omega = Pi

y1[t_] = h0 - m/k*Log[Cosh[t*Sqrt[g*k/m]]]

kk=0.8

V1 = kk*vv[T1] + Sqrt[g*m/k]*Tanh[T1*Sqrt[g*k/m]]

*Mathematica*provides the approximate value of T1 to be 0.4716548296910227 for

*h*

_{0}= 2, we denote it by

*T1*. The velocity \( v(t) = \omega\,\cos (\omega \,t) \) of the floor/racket at

*T1*is \( vv(T1) = V1 \approx 0.279386 \) positive, which indicate that the racquet is moving up. So we add 80% of its velocity to the ball.

On the second stage, the ball bounced up with the initial velocity
\( V1 = kk*vv(T1) + \sqrt{\frac{gm}{k}} \tanh \left( \sqrt{\frac{gk}{m}}\,T1 \right) \approx 4.158 \)
and from the position \( Y1 = \sin \left( \omega\,T1 \right) \approx
0.996038. \) Here *kk* is the coefficients of elastic damping of the racket, which we set to be 0.8.
Therefore, we need to solve two initial value problems:

*T2*is the value of time when v(T2) =0. Then we find ball's position:

*Mathematic*a find its value to be \( T2 \approx 0.846 : \)

T2 = t /. FindRoot[v2[t] == 0, {t, 0.8}]

v3[t_] = -Sqrt[g*m/k]*Tanh[Sqrt[g*k/m]*(t - T2)]

Y2 = y2[T2]

*T3*) when the ball meets the racket:

T3 = t /. FindRoot[y3[t] == Sin[omega*t], {t, 2}]

*t=T2*will be \( y(T2) \approx 1.822 . \) After T2, the ball starts falling down and we need to solve the initial value problems:

*Mathematica*to be sure that the racket does not hit the ball on this stage:

0 < t < T1}, {Y1 + V1*(t - T1) + m/k*Log[Cos[Sqrt[g*k/m]*(T1 - t)]],

T1 < t <= T2}, {Y2 - 4*Log[Cosh[Sqrt[g*k/m]*(t - T2)]], T2 < t <= T3}}]

Plot[{y[t], Sin[omega*t]}, {t, 0, T3}]

*Mathematic*a provides the numerical values:

y4[t_] = Y3 + V3*(t - T3) + m/k*Log[Cos[(T3 - t)*Sqrt[g*k/m]]]

T4 = t /. FindRoot[v4[t] == 0, {t, 2}]

yy4[t_] = YY4 + VV3*(t - TT4) + m/k*Log[Cos[(TT4 - t)*Sqrt[g*k/m]]]

y[t_] = Piecewise[{{h0 - m/k*Log[Cosh[t*Sqrt[g*k/m]]],

0 < t < T1}, {Y1 + V1*(t - T1) + m/k*Log[Cos[Sqrt[g*k/m]*(T1 - t)]],

T1 < t <= T2}, {Y2 - 4*Log[Cosh[Sqrt[g*k/m]*(t - T2)]],

T2 < t <= T3}, {Y3 + V3*(t - T3) +

m/k*Log[Cos[(T3 - t)*Sqrt[g*k/m]]], T3 < t < TT4}, {yy4[t], TT4 < t < 2.081}}]

Plot[{y[t], Sin[omega*t]}, {t, 0, 2.081}, PlotStyle -> {Thick, Thick}]

Claude Perrault | Sir Isaac Newton | Christiaan Huygens |

A tractrix (from the Latin verb trahere "pull, drag"; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a tractor (pulling) point that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed. It is therefore a curve of pursuit. It was first introduced by Claude Perrault in 1670, and later studied by Isaac Newton (1676) and Christiaan Huygens (1692).

Claude Perrault (1613 -- 1688) was a French architect, best known for his participation in the design of the east façade of the Louvre in Paris. He also achieved success as a physician and anatomist, and as an author, who wrote treatises on physics and natural history. Perrault was born and died in Paris. Aside from his influential architecture, he became well known for his translation of the ten books of Vitruvius, the only surviving Roman work on architecture, into French, written at the instigation of Colbert, and published, with Perrault's annotations, in 1673. His treatise on the five classical orders of architecture followed in 1683. As physician and natural philosopher with a medical degree from the University of Paris, Perrault became one of the first members of the French Academy of Sciences when it was founded in 1666.

Sir Isaac Newton (1642 -- 1726/27) was an English mathematician, astronomer, theologian and physicist (described in his own day as a "natural philosopher") who is widely recognized as one of the most influential scientists of all time and a key figure in the scientific revolution. His book Philosophiae Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy"), first published in 1687, laid the foundations of classical mechanics. Newton also made pathbreaking contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

Christiaan Huygens (Latin: Hugenius; 1629 -- 1695) was a prominent Dutch mathematician and scientist. He is known particularly as an astronomer, physicist, probabilist and horologist. Huygens was a leading scientist of his time. His work included early telescopic studies of the rings of Saturn and the discovery of its moon Titan, the invention of the pendulum clock and other investigations in timekeeping. He published major studies of mechanics and optics (having been one of the most influential proponents of the wave theory of light), and pioneered work on games of chance.

**Example**: In order to plot tractrix curve, use the following code:

Manipulate[

ParametricPlot[tractrix[a][t] // Evaluate, {t, 0, .99*\[Pi]},

PlotRange -> {0, 7}], {a, 1, 6}]

Plot[y'[x] = -Sqrt[a^2 - x^2]/x, {x, 0, 20},

PlotRange -> {-10, 10}], {a, 0, 20}]

Plot[-Sqrt[a^2 - x^2] + a Log[a] - a Log[a^2] - a Log[x] +

a Log[a^2 + a Sqrt[a^2 - x^2]], {x, 0, 20}, PlotRange -> All], {a,

1, 20}]

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