## A Sound Correction

As I was taking my coffee over breakfast today, I realized that in the previous post, I did not incorporate the time it took for the sound waves to reach my ear when I pressed the stop button of the stopwatch. Sound waves travel at 340.29 meters per second. At a height of 25 meters, it would take 0.073 seconds for the sound of the explosion to reach the ground. This should be negligible for the purposes of our computation. However, what is really the effect if we incorporate this minor correction?

As seen in the figure below, the recorded time is composed of the time it took for the skyrocket to reach the maximum height plus the time it took for the sound waves to reach my ear, that is,

$t_r = t_a + t_s$

where $t_r$ is the recorded time, $t_a$ is the actual time the skyrocket reached its maximum height and $t_s$ is the time it took for the sound waves to travel from the maximum height to the ground.

We can rewrite the height as

$\displaystyle h_{\text{max}} = - \frac{gt_a^2}{2} + v_i t_a$

Since $v_i = gt_a$,

$\displaystyle h_{\text{max}} = - \frac{gt_a^2}{2} + gt_a^2$
$\displaystyle = \frac{gt_a^2}{2}$

At the maximum height, $h_{\text{max}}$, it would take the sound waves $t_s$ seconds to travel, that is,

$\displaystyle h_{\text{max}} = v_s t_s$

where $v_s$ is the speed of sound. Therefore,

$\displaystyle t_sv_s = \frac{gt_a^2}{2}$
$\displaystyle (t_r- t_a)v_s = \frac{gt_a^2}{2}$
$\displaystyle v_st_r - v_st_a = \frac{gt_a^2}{2}$
$\displaystyle \frac{gt_a^2}{2} + v_st_a - v_st_r = 0$

The last equation above is a quadratic equation which we can solve for $t_a$ using the quadratic formula:

$\displaystyle t_a = \frac{-v_s \pm \sqrt{v_s^2 - 4(1/2)gv_st_r}}{2(1/2)g}$

Substituting the values $g = 9.8$, $v_s=340.29$, and $t_r = 2.3$, we get

$\displaystyle t_a = 2.22$

Using this result, the maximum height is computed to be

$\displaystyle h_{\text{max}} = \frac{gt_a^2}{2} = 24.33$

Comparing this with our previous computation of $h_{\text{max}} = 25.829$, we find that we have overestimated the maximum height by about 1.49 meters. It’s not really that bad so we can just drop the effect of sound.

## New Year, New Heights and the Projectile

I spent my new year celebration at our friend’s place with 2 other colleagues. It was started with a dinner at about 9:30 pm and we bought some bottles of wine in addition to the spirits we already brought. At about 10:30, a street party was setup in the vicinity and about 5 other families joined in a potluck. Marijo’s brother started the fireworks at about that time with David and Gerard (my colleagues) taking turns setting them off. I did not attempt to set them off because I did a lot those things way back as a child and still lucky enough to have my fingers intact.

This is a video of our 2012 New Year’s Celebration:

As Gerard was flying off skyrockets (or kwitis as they are called in my language) David asked me “How high do you think these things go up?”. That was a really good question which got me to thinking. I said “Let’s compute!” So I took my iphone and launched its built-in stopwatch. Gerard flew about 3 skyrockets and I took the time it takes for them to fly until they blow up in the air. The average was found to be 2.3 seconds.

Independent of Mass

Legend has it that Galileo Galilei performed an experiment at the Leaning Tower of Pisa. He dropped objects of different masses to see which of them falls first. Starting from the heaviest to the lightest object, what he found was that, in a vacuum, they reach the ground at the same length of time when dropped from the same height. Using this result, we can compute for the distance travelled by any object under the influence of gravity. At heights near the earth’s surface, the acceleration of objects in the presence of gravity alone is given by

$\displaystyle \frac{d^2h}{dt^2}= \frac{dv}{dt} = -g$

where $h$ is the vertical distance (height), $v$ is the velocity and $g$ is the acceleration due to gravity given by $g = 9.8 \text{m/s}^2$.

The equation above is a simple differential equation which we can solve for the velocity at any time $t$ by integrating both sides:

$\displaystyle \int^{v}_{v_i} dv = \int^{t}_{t_i} -g dt$

which by the Fundamental Theorem of Calculus gives us

$\displaystyle v - v_i = -g(t - t_i)$
$\displaystyle v = -gt + v_i + t_i$

If we take initial time of the rocket launch to be zero, that is, $t_i=0$, we have

$\displaystyle v = \frac{dh}{dt}= -gt + v_i$

We don’t know the initial velocity $v_i$ of the rocket. However, at the maximum height attained by the rocket, the velocity is zero. Using this fact, we can solve for the initial velocity:

$\displaystyle 0 = -gt_{\text{max}} + v_i$
$\displaystyle v_i = gt_{\text{max}}$

where $t_{\text{max}}$ is the time the rocket reached maximum height.

At $t_{\text{max}}=2.3$ seconds, the initial velocity is therefore $v_i = 9.8 * 2.3 = 22.5$ m/s.

Maximum height

We can solve for the height as a function of time by integrating both sides of the velocity equation:

$\displaystyle \int^{h}_{h_i} dh = \int^{t}_{t_i} -gt dt + v_i dt$
$\displaystyle h - h_i = -\frac{gt^2}{2} + v_i t \big|^t_{t_i}$

Since the initial height is zero , that is , $h_i = 0$ and the initial time is zero $t_i = 0$, we have

$\displaystyle h = -\frac{gt^2}{2} + v_i t$

Since at t=2.3 seconds, the rocket reaches the maximum height, we have

$\displaystyle h_{\text{max}} = -\frac{9.8 \times (2.3)^2}{2} + 22.5 * 2.3 = 25.829 \text{ meters}$

Does this make sense? The average height of a building storey is about 3 meters. This is about 8 or 9 storeys high which I think makes sense.

It’s good that my new year not only started with a warm welcome from friends but it also appealed to my physics curiosity.

Thank you to Marijo Condes and family for letting us spend a memorable New Year’s Eve.

## Topology and the Causality of Spacetime

While preparing for tomorrow’s first day of office for 2011, I stumbled upon a hard-copy of my undergraduate thesis. I realized that I don’t have any backup of this piece of work. Thanks to the JotNot application of iPhone, I was able to scan my thesis and convert it to PDF.

Below are samples of the scanned pages. Click on each thumbnail to get the full resolution.

So, if you want to understand how Stephen Hawking and Roger Penrose arrived at their Singularity Theorems, you can read it from the PDF version of my thesis:

Topology And The Causality Of Spacetime

N.B., the file is 40 MB so have some patience.

Einstein’s Theory of General Relativity is one ofthe greatest intellectual achievements in this century. It radically changed our way of looking at the world. lt particularly freed us from our usual Euclidean way of looking at the universe, However, even without the use of Einstein’s equation we can already say something about the universe we live in. It has something to do with the manifold structure of spacetime and the causality conditions imposed bythe speed of light, in that no signal could travel at speeds greater than the speed of light. At this topological level, we impose certain conditions that our spacetime should posses in order for it to be called a reasonable model of our universe.

This thesis is a review article of the causal structure of spacetime. By causal structure, we mean those properties a spacetime should possess so that two pairs of points in it can be joined by a causal curve. A basic notion such as this can already give many results in terms of our understanding of the universe.

When we say that a given model of the universe is reasonable , we mean the following: (i) a lorentz metric can be defined on it,(ii) it shoud be orientable in space and time, (iii) it does not contain closed timelike curves and those curves, though not closed but are “almost closed” , which will be made more precise later, (iii) it possesses a surface( achronal) where we can define initial data set that will determine the evolution of the universe. `

In Chapter ll, we are going to review those aspects of mathematics that are useful in causal analysis. We will start with a review of manifold theory. We will define what we mean by a manifold and all other defined structures on it. We will notice that even without the metric, many consequences of the conditions enumerated above will be derived The only thing we assume on our manifold is that it allows the existence of a Lorentz metric.

In Chapter III, we will show that not all manifolds admit a Lorentz metric. In fact it does not make sense to talk of the surface of the sphere as a two dimensional model of spcetime since a Lorentz metric cannot be defined on the sphere. We will also show that compact spacetimes are of little interest to cosmology because they allow the existence of closed timelike curves.

Chapter IV is a brief chapter that applies the concepts introduced in Chapter III to the proof of Hawking’s Singularity Theorem. Here we will define what we mean by a singularity and learn that a generic class of solutions of Einstein’s equation is singular. Singularities do not tell us that the theory itself is defective, but that they are really physical entities that exist.

Although causal analysis is already a mature subject, it has only been available to a handful of people because it’s nature is in a way abstract. The purpose therefore of this review is to present more details of the subject and providing proofs to some statements that have been omitted in standards texts.

## John Wheeler, A great physicist!

John Archibald Wheeler is one of my most admired physicist. He just died recently and I want to make a small tribute to him to did so much in General Relativity. Wheeler is the person who coined the word “black hole”, a concept which captivated my imagination when I was yet a kid and a concept which I eventually studied on my own in the University. He wrote one of the best books in General Relativity entitled “Gravitation”. This is a thick and heavy book full of physics and mathematics. It introduces advanced mathematical concepts as you progress in your study of Relativity in a very geometric way. This book has full of illustrations that really whets your appetite for studying advanced physics.

I have bought many books authored by Wheeler. One of them is “Exploring Black Holes“, with Taylor as co-author. I bought this book in Borders bookstore in Singapore. Unfortunately, we don’t have these kinds of books in the Philippines. A friend of mine also lent me a layman’s book written by Wheeler entitled “Geons, Black Holes & Quantum Foam”. This book gave accounts on Wheeler’s contribution to the Manhattan Project and the other great people whom he worked with.

I have a professor before who took his PhD in US who told me that whenever Wheeler gives a lecture at 7 am in the morning, he will always attend it no matter how cold it is in New York. That’s how big an impact Wheeler had on him.

To the great man Wheeler, thank you for inspiring us to study one of the greatest theories in the 20th century!