Cannons and Space ships

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Further reading

Katherine Johnson, computer at Lanley Labs for NASA
Katherine Johnson, computer at Lanley Labs for NASA

Friendship 7

In February 1962, Katherine Johnson sat in the Langley Aeronautics Laboratory in Langley, Virgina, busily scribbling away. She was in the middle of a length calculation that was a crucial part of an up-coming mission.

Katherine was one of an elite cadre of African American computers working for NASA, and featured in the 2016 movie Hidden figures. Prior to the advent of the digital computer, computers were people hired to perform the long calculations needed in science and engineering projects. Despite systematic discrimination against women and African Americans at this time, computing positions offered a path for advancement. NASA’s calculation needs were often new and complicated, sometimes requiring mathematical understanding beyond that typically gained with undergraduate degrees.

On this day, Katherine Johnson was calculating the solution of a system of nonlinear differential equations (by hand!) that would predict the landing of the United States of America’s first manned orbital flight – John Glenn’s flight in the Mercury Friendship 7 space capsule. The calculation had already been performed twice – once by hand and once by a new IBM digital computer program – with different answers each time. Katherine had previously worked on the orbital calculations for Alan Sheperd’s flight, and Glenn had ask for her help in determining which was right.

To put perspective on the challenges and depth of the calculations Johnson and her colleagues performed, lets trace the story of her calculation through our long fascination with projectile motion.

Cannons in the Renaissance

Daniel Santbech’s ballistics diagram
Daniel Santbech’s ballistics diagram

Since apes began hunting with stones and spears, we have sought to understand the motion of thrown objects. Today, even when our lives seem distant from those of our ancestors, children still love to play catch. While we can develop quite good intuitive understandings of ballistic motion, precise predictive descriptions did not start to appear until the cannon became a decisive instrument of war, when the Byzantine Empire fell to the Ottoman Empire in 1453.

Mathematics was reborn as a tool for the study of the natural sciences during the European Renaissance. Thanks in part to Leonardo Pisano Fibonacci’s appropriation of the Hindu place-value arithmetic in the 1200s, calculation had already shown its worth and become an essential part of commerce. By the late 1500s, mathematics was taught in the best schools around the continent. Math curricula relied on texts from the classical era, including Euclid on Geometry and Optics, Apollonius’s Conics, and Ptolemy’s Almagest and Geography. Scholars soon discovered errors and gaps in the classical texts that needed filling. As Renaissance mathematicians expounded on the classics, they also developed mathematics in new directions for personal and financial interests. One of these new directions was artillery training.

As a practical problem, the ranging of a cannon was an involved process. The range depended on the bore of the cannon, the weight and shape of the cannon ball, the lay of the land, the strength of the wind, the amount of gunpowder used, and the angle of firing. Trial-and-error could lead to good targeting, but with each firing, recoil might necessitate re-targeting and ammunition cost money. In cases where time was brief or power and cannon balls were in short supply, accurate ranging could make an important difference in a battle.

In the 1300s, Jean Buridan his contemporaries Nicole Oresme and Alberto Saxony had reconsidered the conventional philosophical explanation of falling bodies put forward by Aristotle. Aristotle had distinguished between weak and violent motions. Weak motions, exemplified by smoke rising and leaves falling, were where the medium of the world resisted the motion. Violent motions like oars passing through water were where the medium perpetuated the motion. Oresme countered that the medium always imposed drag on motion, and that the tendency of objects to continue moving was a consequence of their initial impetus rather than the medium. Albert of Saxony proposed a 3-stage description of projectile flight – a diagonal line up, a curve near the top where drag slows the flight, and a straight line down at the end. This was based on two general observations. First, when a projectile is launched into a near-by barrier, it appears to travel in roughly a straight line based on the angle of launch. Second, if the projectile is not pre-maturely stopped, then after being slowed by drag, the final part of a projectile’s flight path will descend straight down.

A major step forward in projectile theory was taken by Niccolo “Tartaglia” Fontana, a Venetian mathematician famous for his solution of the cubic equation. In 1537, printers published a book by Tartaglia on ballistics intended for the use of artillerymen. It was the first book of its kind. Tartaglia used a circular arc to fill in Albert’s intermediate stage, providing the first complete quantitative models for a ballistic trajectory. In the course of his analysis, Tartaglia conjectured that the greatest range of an artillery piece would be achieved when firing at a 45 degree angle.

Tartaglia’s ballistics diagram from 1537

Galileo Galilei and his gradstudent Evangelista Torricelli took a different approach in 1638. Aristotle had argued that heavier bodies fall faster than light bodies, and this had been widely accepted through the 15th century.
Galileo, however, questioned this conclusion. Based on some thought experiments and some actual experiments with rolling balls, Galileo came to believe that falling objects accelerate at the same rate, no matter what their weight. If a piece of an object wanted to fall more slowly than the whole, wouldn’t the object pull itself apart? Logical pursuit of this idea leads to the prediction that the trajectory of a projectile should be a parabola.

Empirically, we’ve long observed that gravity has about the same effect on us, whereever we are in the world; if we’re at the beach or a mountain top, in a boat or hot-air balloon, how hard it is to lift things and how high we can jump doesn’t change very much. Thus, gravity seems like a constant, universal thing. If we assume gravity exerts an approximately constant acceleration, represented by the symbol \(g\), on all objects, then the height of a falling body \(y(t)\) as a function of time \(t\) is given by \[y(t) = \frac{g t^2}{2} + \dot{y}(0) t + y(0)\] where \(y(0)\) is the height at time \(t=0\) and \(\dot{y}(0)\) is the velocity at time \(t=0\). Measurements of projectile motion have found the gravitational acceleration on earth is \(g = 9.8 \; \text{m}/\text{s}^2\).

[ Data : hide , shown as table , shown as CSV shown as QR ]

# x-position, y-position, shock?
# Extracted with g3data by T. Reluga, 2016-09-24 from Berenice Abbott's bouncing ball timelapse photograph
# Points were extracted in order, so assuming equal time between strobes, the position represents time
# The data are overly precises -- the hand accuracy has large error bars, may 50% of golfball radius
0.2043596,0.017391, # bounce 1
0.5395095,0.020289, # bounce 2
0.7929155,0.043478, # bounce 3
0.9727520,0.028985, # bounce 4

Like Tartaglia’s model of ballistic motion, Galileo’s is an idealization. Depending on how we personally perceive scientific truth, Galileo was at best only close to being right some of the time, and at worst, his model is wrong. For example, it is easy to test for ourselves that there are circumstances where Aristotle was correct and heavier objects fall faster – dropping objects of different weights into a fish tank, for example. Both Tartaglia and Galileo had empirical evidence to support their theories (though Galileo may deserve more credit for conducting his own experiments). Those attempting to apply Galileo’s theory to the practice of ballistics were often frustrated. For example, a Genoese man named G. B. Renieri, wrote to Galileo’s student Torricelli in the mid 1600’s to complain about the practical failure of parabolic projectile theory (Segre, 1983).

The triumph of Galileo’s theory was ultimately not because of its immediate practical improvements, but in its inspirational value, opening the roads that would eventually lead to Newton’s laws and more powerful descriptions of ballistic motion that would be more accurate.

Newton’s laws of motion

Nature and Nature’s Laws lay hid in Night:
God said, “Let Newton be!” and all was light. – Alexander Pope (1688–1744)

These two lines capture the essence of the story of the fusion of mathematics and natural philosophy that gave birth to the modern science of physics. In some tellings, the story is boiled down still further to a single “eureka!” moment in Isaac Newton’s life – the fall of an apple from a garden tree as the young Newton looked on. While amusing and easy to repeat, this is about as close to the truth as our typical contemporary Hollywood retelling of history. Over two centuries, numerous scholars’ studies built up a tinder pile whose ignition was the start of the age of enlightenment. The fire was sparked by the mixing of the study of motion with the mathematics we now call calculus. Curious folk like Tartaglia, Galileo, Snell, Kepler, Descarte, Wren, Hooke, Hugyens, and others had been investigating the mechanics of motion. In the meantime, even before the writings of Gottfried Leibniz and Isaac Newton, calculus had been developing in the works of Cavalieri, Kepler, Pascal, Roberval, Wallis, Barrow, and others. Even though all of these advancements were being made, it is widely agreed that Newton’s Principia Mathematica published in 1687, is the combustion event when these two areas of studies were first brought together. Amazingly, Principia itself makes no mention of the new mathematics of calculus. Newton’s contemporaries and successors were left to flesh out the language of the mechanics of calculus.

In Principia Mathematica, Isaac Newton proposed three laws for the description of natural motions.

  1. A body at rest stays at rest, and a body in motion stays in motion until acted on by a force.
  2. Any change in velocity must be proportional to and in the direction of the total force applied to it.
  3. For every action, there is an equal and opposite reaction.

Each of the three laws is an aspect of the more general law of conservation of momentum – the momentum of each part of a system is conserved until external forces act on it. Thus, to describe an objects state, one needs to specify not just the objects position, but also its mass and velocity. Leonhard Euler gave Newton’s first two laws their modern mathematical form \[\begin{gather} m \frac{d^2\vec{x}}{dt^2} = \sum _ {i} \vec{F} _ i \end{gather}\] where \(m\) is an objects mass, \(\vec{x}(t)\) is the position vector, and each \(\vec{F} _ i\) is an externally applied force. For example, Galileo’s formula for simple vertical ballistic motion can be written using Euler’s formula as \[m \ddot{y} = m g\] where \(mg\) is the downward force of gravity, \(g\) is the acceleration of gravity, \(y(t)\) is the height, and Newton’s dot-notation is used as as shorthand for the second derivative of with respect to time, It is a credit to Isaac Newton’s clarity of thinking that his laws so naturally translate to Euler’s formula despite his avoidance of calculus and his ignorance of modern vector algebra.


Newton’s laws give us a clear method to extend Galileo’s ballistic motion model to include the effects of drag that earlier philosophers like Tartaglia had recognized as important. A projectile has two forces acting on it: a constant gravitational force pulling it down and the force of drag (a.k.a. air resistance) acting opposite to the direction of motion. As the speed increases, the force of drag also increases. But without more information, we don’t know the concrete nature of this relationship. One hypothesis is that the force of drag is a power-law function of the speed. Let’s explore this hypothesis.

To put these into an equation, let \(\vec{z}(t) = x + y i\) be the position vector of the projectile, and \(\vec{v}(t) = v_x + v_y i\) be the velocity vector. Applying Euler’s formula for Newton’s laws, we get the following sytem of ordinary differential equations for \(\vec{z}\) and \(\vec{v}\). \[\begin{align} \frac{d \vec{z}}{dt} &= \vec{v}, \\ m \frac{d \vec{v}}{dt} &= - m g i - k |\vec{v}|^{n-1} \vec{v}, \end{align}\] where \(|\vec{v}| = \sqrt{ v _ x ^2 + v _ y ^2 }\). Although we can not tell from just looking at this system, it turns out to be helpful to transform the velocity equation from Cartesian to polar coordinates. Let us rewrite the Cartesian velocity vector \(\vec{v} = v_x + i v_y\) in terms of its speed \(v\) and heading \(\gamma\) as \[\vec{v}(t) = v(t) e^{i \gamma(t)} = v(t) ( \cos \gamma(t) + i \sin \gamma(t))\] For the heading, \[\begin{gather} \tan \gamma = \frac{v_y}{v_x} \\ (1 + \tan^2 \gamma ) \dot{\gamma} = \frac{v_x \dot{v}_y - v_y \dot{v}_x}{v_x^2} \\ (1 + \frac{v_y^2}{v_x^2} ) \dot{\gamma} = \frac{v_x \dot{v}_y - v_y \dot{v}_x}{v_x^2} \\ \dot{\gamma} = \frac{v _ x \dot{v} _ y - v _ y \dot{v} _ x}{v^2} \\ v \dot{\gamma} = \dot{v} _ y \cos \gamma - \dot{v} _ x \sin \gamma \\ v \dot{\gamma} = (-g - k/m v^n \sin \gamma) \cos \gamma - ( - k/m v^n \cos \gamma ) \sin \gamma \\ v \dot{\gamma} = -g \cos \gamma \end{gather}\] Then, assuming the speed is always positive, \[\begin{align} m \frac{d v}{dt} &= -m g \sin \gamma - k v^n, \\ v \frac{d \gamma}{dt} &= -g \cos \gamma. \end{align}\]

This nonlinear system for speed \(v\) and heading \(\gamma\) can tell us a few things. When we look for steady-state solutions, we see the heading will not change (\(d \gamma / dt = 0\)) once it is pointing straight up (\(\gamma = \pi/2\)) or down (\(\gamma = -\pi/2\)). The speed will also be steady (\(d v/dt = 0\)) when \(m g \sin \gamma = - k v^n\). There is no positive solution of this equation when the projectile is going straight up (\(\gamma = \pi/2\)), but there is when the projectile is falling straight down (\(\gamma = -\pi/2\)). When the projectile is falling straight down, there is a steady-state velocity (called the “terminal velocity”, \(v _ T\)) where drag forces balance gravitational forces. The terminal velocity is given by the formula \[v _ T = \sqrt[n]{\frac{m g}{k} }.\]

The terminal velocity formula turns out to be very useful because it establishes a scaling relationship that depends on the mass of the falling object. The drag coefficient \(k\) is different for each object, but scaling exponent \(n\) is the same over a wide range of objects and conditions. By doing a simple experiment, we can estimate the general value of the scaling exponent. Perhaps the simplest way to do this these days is by using coffee filter papers and a video camera or range finder. The coffee filters are very light, so they fall slowly, and speed is easy to measure because their upside-down parachute-shape keeps them stable as they fall.

Plot of coffee filter experiment showing terminal velocity is roughly proportional to the square root of mass
Plot of coffee filter experiment showing terminal velocity is roughly proportional to the square root of mass

[ Data : hide , shown as table , shown as CSV shown as QR ]

# num of coffee filters, terminal velocity (m/s)
# The following is a data set collected by students
# who observed the terminal velocities of nested coffee
# filters falling in a stable way using an ultrasonic
# range finder and a little post-processing. Each 
# case was replicated 6 times.
# Although we don't know the mass of the filters,
# we can still use the data to determine how the
# terminal velocity scales with mass.
# column 1 - number of nested coffee filters
# column 2 - terminal velocity in meters per second
9, 2.314
9, 2.476
9, 1.994
9, 2.306
9, 2.345
9, 2.313
8, 2.244
8, 2.299
8, 2.258
8, 2.241
8, 2.141
8, 2.309
7, 2.097
7, 1.722
7, 2.093
7, 1.986
7, 1.932
7, 2.104
6, 1.988
6, 2.027
6, 2.019
6, 1.908
6, 1.829
6, 2.044
5, 1.854
5, 1.821
5, 1.792
5, 1.899
5, 1.681
5, 1.801
4, 1.599
4, 1.673
4, 1.739
4, 1.677
4, 1.700
4, 1.710
3, 1.321
3, 1.379
3, 1.537
3, 1.501
3, 1.412
3, 1.401
2, 1.121
2, 1.088
2, 1.187
2, 0.935
2, 1.273
2, 0.971
1, 0.790
1, 0.817
1, 0.811
1, 0.850
1, 0.766
1, 0.754

These experiments lead to the conclusion that drag force on a body is proportional to the square of the speed of the body’s motion. The final form for the equations of motion is then \[\begin{align} m \frac{d v}{dt} &= -m g \sin \gamma - k v^2, \\ v \frac{d \gamma}{dt} &= -g \cos \gamma, \\ \frac{d z}{dt} &= v e^{i \gamma}. \end{align}\] The task left to us is to solve this system – not an easy task. Newton erred in his own analysis in the first edition of Principia, and the error was pointed out to him by Johann Bernoulli, leading to awkward correction of the second addition and contributing to his legendary dispute with his continental peers.

Johann’s full solution of the equations of motion can be written in terms of “quadratures”, which means we’re allowed to include unevaluated integrals. Since the angle \(\gamma(t)\) changes monotonely with time, we can use it as our independent variable instead of time, so \[\begin{gather} \frac{d v}{dt} = \frac{d v}{d\gamma} \frac{d \gamma}{dt} = \frac{d v}{d\gamma} \left( \frac{g \cos \gamma}{v} \right). \end{gather}\] Substituting into the acceleration equation and simplifying, \[\begin{gather} \frac{d v}{d\gamma} \cos \gamma - v \sin \gamma = - \frac{k}{mg} v^{3}. \end{gather}\] Johan recognized that this is one of a large class of nonlinear equations – a Bernoulli equation – discovered by his brother that could be solved by a clever and relatively simple substitution, leading to a complete solution written in terms of integrals (see Exercises below).

Landing a spaceship

In February 1962, when Katherine Johnson was working on her calculations for the trajectory of Friendship 7, Bernoulli’s cannonball equations were the natural place to start. But the scales involved in the landing of a spaceship compared to shooting a cannonball were so much larger that the equations had to be re-framed. In 1964, she and one of the engineers published a paper on approximating the flight path reentry angle of a spaceship, based on these re-framed equations.

equations for spaceship reentry

First, the position of the craft can no longer be treated in a rectangular coordinate system – rectangular coordinates can not describe orbits around the earth. Instead, we can use polar coordinates. Let \(r\) be the approximate radius of the earth, \(h\) be the altitude of the craft above the ground, and \(\theta\) be the angular position of the craft. Thus, the position of the spacecraft (in modified polar coordinates) changes according to the system \[\begin{align} \frac{dh}{dt} &= v \sin \gamma \\ (r + h) \frac{d\theta}{dt} &= v \cos \gamma \end{align}\] Another modification is needed because of the use of polar coordinates. In rectangular coordinates, the heading angle \(\gamma\) is changed only by the action of gravity accelerating the spaceship downward. However, in polar coordinates, the heading angle will continue to change even in the absence of gravity. It is natural to assume the heading angle is always measured relative to nearest point on the earth’s surface. When there are no forces acting on the spaceship, it will travel in a straight line. The heading angle, then, between the surface of the earth will change at the same rate as the angular position \(\theta\) (see exercises). \[\begin{gather} \frac{d\gamma}{dt} = \frac{d\theta}{dt} = \frac{v}{r + h} \cos \gamma \end{gather}\] If we now include the effect of gravity on heading, as Bernoulli did, \[\begin{align} v \frac{d\gamma}{dt} &= \frac{v^2}{r+h} \cos \gamma - g \cos \gamma. \end{align}\] Third, the drag force is still proportional to the square of the velocity, but is also proportional to the density of the air, and the density of air varies dramatically between the surface of the earth and space. As was previously found in one of our scaling problems, the density of the atmosphere decreases exponentially with altitude \(h\), at least in the range of the lower stratosphere. Thus, \[\begin{align} m \frac{dv}{dt} &= - m g \sin \gamma - k v^2 e^{-B h} \\ v \frac{d\gamma}{dt} &= k v^2 e^{-Bh} + \frac{v^2}{r+h} \cos \gamma - g \cos \gamma \end{align}\] Finally, earth’s atmosphere is very thin relative to the size of the earth – the main part of earth’s atmosphere, the troposphere, is like a pillowcase on a basketball. When \(r \gg h\) over the domain of interest, it was common practice to take \(r + h \approx r\) anyplace the sum appeared in the system. All together, \[\begin{align} m \frac{dv}{dt} &= - m g \sin \gamma - k v^2 e^{-B h} \\ v \frac{d\gamma}{dt} &= k v^2 e^{-Bh} + \frac{v^2}{r} \cos \gamma - g \cos \gamma \\ \frac{dh}{dt} &= v \sin \gamma \\ r \frac{d\theta}{dt} &= v \cos \gamma \end{align}\]


  1. When a body falls from rest, we can see that its speed increases as it falls. But before the work of Galileo, this acceleration was incorrectly understood. The speed-distance theory, supported by some in the Jesuit order of Catholic priests, was that as a body falls, its speed is proportional to the distance it has travelled. Thus, the greater the distance fallen, the faster the body. Mathematically, \[\frac{y'(t + \Delta t)}{y'(t)} = \frac{y(t + \Delta t)}{y(t)}.\] Pierre Fermat (famous for his work in number theory) demolished this speed-distance theory by showing that it contradicts reality.
    1. Derive a second-order differential equation for the speed-distance theory as \(\Delta t \rightarrow 0\).
    2. Solve the speed-distance differential equation for \(y(t)\).
    3. Discuss what happens with the solutions of the speed-distance equation in the special case of a body falling from rest (\(y'(0) = 0\)).
  2. In Two New Sciences, Galileo argued that the speed of a falling body was proportional to the time for which it had been falling. Mathematically, \[\frac{y'(t + \Delta t)}{y'(t)} = \frac{t + \Delta t}{t}.\] Galileo thus placed time at the center of the theory of motion for the first time.

    1. Derive a second-order differential equation for \(y(t)\) as \(\Delta t \rightarrow 0\).
    2. Solve this differential equation for \(y(t)\).
  3. (The Newton–Bernoulli problem) An ordinary differential equation for an unknown function \(z(t)\) is called a Bernoulli equation in honor of Jacob Bernoulli if it can be written in the form \[\frac{dz}{dt} + p(t) z = g(t) z^n.\] Bernoulli equations can be transformed into linear differential equations with the transformation \(u = z^{1-n}\). Use this transformation to complete the integration of Johan Bernoulli’s equations for ballistic motion under air resistance. \[\begin{gather} \frac{d v}{d\theta} \cos \theta - v \sin \theta = - \frac{k}{mg} v^{n+1}. \end{gather}\]

  4. In class, we used the principle of conservation of momentum to derive the following 2nd-order nonlinear system of equations for the motion of a returning space capsule, with Johnson’s correction for air-resistance, \[\begin{gathered} m \begin{bmatrix} \ddot{x} \\ \ddot{y} \end{bmatrix} = -\frac{G m M}{(x^2 + y^2)^{3/2}} \begin{bmatrix} x \\ y \end{bmatrix} - k e^{-B [ (x^2 + y^2)^{1/2} - R]} \sqrt{\dot{x}^2 + \dot{y}^2} \begin{bmatrix} \dot{x} \\ \dot{y} \end{bmatrix}\end{gathered}\] where \(M=5.972 \times 10^{24}\) kilograms is the mass of the earth, \(G = 6.674 \times 10^{-11}\) is the universal gravitational constant, the radius of the earth \(R = 6.378 \times 10^6\) meters, the mass of the capsule \(m = 1,200\) kilograms, the drag coefficient \(k = 8.33 \times 10^{-8}\) and the density decay rate \(B= 4.8 \times 10^{-5}\) per meter. We will assume that the earth is perfectly circular around the equator. All constants and variables should have dimensions expressed in terms of the fundamental units of kilograms, meters, and seconds.

    1. Rewrite our 2-d 2nd order system as a 4-d first-order system, using \(\dot{x} = u\) and \(\dot{y}=v\).

    2. If a space capsule at the Karman jurisdictional line of 100 kilometers is travelling around the equator with an initial speed of \(7.85 \times 10^3\) meters per second tangent to the earth’s surface. Plot the earth’s surface and the x-y coordinates of the space capsule from the initial time until touch-down.

    3. How many degrees of longitude will the capsule travel before landing? Give your answer to 2 decimal places.

  5. Below is a small data set on cannon range as a function of gun elevation (angle of firing above horizontal) extracted from H. Douglas’s 1855 edition of A Treatise on Naval Gunnery. Use these data to infer the muzzle velocity and drag coefficient for the cannon. needs testing

[ Data : hide , shown as table , shown as CSV shown as QR ]

# Elevation (degrees), Measured Range (yards) 
# Measured range of how far cannon balls went
# when fired from a cannon with different elevation
# angles.
# 42 lb round shot with 14 lb powder charge
# predicted muzzle velocity of 1684 ft/sec
0 400
1 1045
2 1263
3 1622
4 1770
5 1938
6 2100
7 2300
8 2580
9 2650
10 2900
  1. TO DO It would be great to have a data set for actual long-range projectile motion that could be compared over time with the theory. Can anybody find such a data set on the web?

  2. Rockets
    1. Without using any equations, describe main events in the course of the flight of a rocket with a single-stage solid-state engine launched from the ground up into the atmosphere.
    2. State 3 assumptions that you will make in setting up your equations, and conditions under which they are not reasonable assumptions.
    3. What are two important differences between the motion of a rocket and the motion of a projectile?
    4. Derive a set of differential equations for the vertical motion of a rocket.
  3. Use Euclidean geometry to show that, in the absence of all forces acting on a spaceship, the rate of change of the heading angle \(\gamma\) is the same as the rate of change of the angular position \(\theta\).

  4. Consider the following equation accounting for the change in gravity with altitude. \[\frac{d^2y}{dt^2} = - \frac{g R^2}{(y + R)^2}, y(0)=0, \dot{y}(0) = v\] where \(R\) is the radius of planet, \(g\) is the acceleration of gravity at the surface, and \(v\) is the initial upward velocity.

    1. Show using dimensional analysis that this equation can be rewritten as \(\ddot{x} = -(1+\epsilon x)^{-2}, x(0) = 0, \dot{x}(0) = 1\).
    2. Approximate using regular perturbation methods (series, parametric differentiation, or successive approximation)
  5. Check Galileo’s data on projectile motion (see Drake, 1973).

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