Did you think Olaf would forget that you remain a disgrace to the human species?!
Olaf is here to make your existence marginally less useless!
We have already seen velocity defined as $\frac{x_1 - x_2}{t_1 - t_2}$ as $t_1 - t_2$ becomes smaller and smaller.
We shall now present the necessary mathematics in order to make such a wishy washy wooly and wimpy statement rigorous enough to withstand a barbarian's bachelor's party!
We have already discussed the real numbers $\mathbb{R}$. Today I will introduce the subset $\mathbb{N} \subset \mathbb{R}$ defined by:
$\mathbb{N} = \left\{1,2,3,4,..\right\}$
We shall refer to $\mathbb{N}$ as the set of natural numbers.
Notice that $n \in \mathbb{N} \implies n + 1 \in \mathbb{N}$. This leads to a rich application in mathematics, computer science and mating, as it is the underlying fact behind the techniques of mathematical induction and recursion. However, these will not detain us at this stage.
Consider now an infinite sequence of real numbers, labbeled: $a_1, a_2, a_3,...,a_n,...$. For convenience, the entire sequence may be referred to as $(a_n)_{n \in \mathbb{N}}$
consider the sequence defined by $a_n = \frac{1}{n}, n\in \mathbb{N}$
As $n$ is chosen to be larger and larger, $\frac{1}{n}$ will become smaller.
Indeed, for any real number $\epsilon > 0$, I can find an $N \in \mathbb{N}$ such that $\frac{1}{N} \leq \epsilon$ In this sense, we say that the sequence $(\frac{1}{n})_{n\in \mathbb{N}}$ appraches 0 as $n$ approaches infinity.
We shall generelise this notion, giving the definition of a limit of a sequence:
A sequence $(a_n)_{n\in \mathbb{N}}$ is said to converge to $a$ if for every $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that $|a_n - a| \leq \epsilon$ for all $n \geq N$. $a$ is also known as the limit of the sequence.
We can write this as $\underset{n \rightarrow \infty}{\lim} a_n = a$ or $a_n \underset{n \rightarrow \infty}{\longrightarrow} a$.
if $(b_n)_{n \in \mathbb{N}}$ is some other sequence which converges to $b$, then then we have the following result, whose proof would be up to you to produce! FOR ALL BARBARIANS PRODUCE THEIR OWN PROOFS!
$\underset{n \rightarrow \infty} {\lim} (a_n + b_n) = \underset{n \rightarrow \infty} {\lim} a_n + \underset{n \rightarrow \infty} {\lim} b_n = a+b$
$\underset{n \rightarrow \infty} {\lim} (a_n \times b_n) = \underset{n \rightarrow \infty} {\lim} a_n \times \underset{n \rightarrow \infty} {\lim} b_n = ab$
$\underset{n \rightarrow \infty} {\lim} (\frac{a_n}{b_n}) = \frac{\underset{n \rightarrow \infty} {\lim} a_n} {\underset{n \rightarrow \infty} {\lim} b_n} = \frac{a}{b}$ if $b \neq 0$
We are now equipped with the proper weapons to attack calculus!
But you now require rest. GIVE ME 1000 PUSHUPS NOW, WEAKLINGS
Thursday, February 3, 2011
Tuesday, February 1, 2011
Momentum, or, Why I will always beat you
At the risk of causing injury to you, I will now introduce some physics. If you find yourself unable to lift the required weight; resign yourself to a life of tedium and serfdom, for the free and great life of a barbarian is far beyond your abilities.
Every barbarian must have an acute understanding of physics, be it while fighting in glorious hand to hand combat, sailing your longship to ripe pillaging fields or calculating the trajectory a system undergoes through state space during a scattering process whilst remaining Lorentz -covariant.
Without being marred with philosophy, I shall make a few statements which are self evident (AGAIN, LOGIC!) and then construct the foundations of what may be called "Newtonian Mechanics", after one of the great barbarians: Newton the Madaxe (although most sources erroneously call him Isaac Newton)
Space can be considered as an ordered collection of numbers $(x,y,z)$ such that $x,y,z \in \mathbb{R}$. Upon fixing the 3 numbers representing some specific location in space, such as yourself, all other points may be unambiguously defined.
A simple way to do this systematically is to decide on a left and right, forward and back, up and down while the numbers $x,y$ and $z$ would correspond to a distance from the initial fixed point along those directions.
We can now know where everything is! However, bitter experience has shown us that things do not necessarily remain where they are right now. So therefore, we require a measure of time, $t \in \mathbb{R}$.
We also require another quantity mass, $m \in \mathbb{R}$, which encompasses "how much" of something there is.
From these we can extract the necessary physics.
An object of mass $m$ can therefore be labelled by a succession of different positions $(x,y,z)$ for every time $t$, with time constantly increasing.
Define the velocity $\bold{v} = (\dot{x}, \dot{y}, \dot{z})$ of an object by the change of positions divided by the change in times, as the change in time goes smaller and smaller.
In other words:
$\dot{x} = \frac{x_1 - x_2}{t_1 - t_2}$ where $x_1$ is the value of $x$ at time $t_1$, and so on, while $t_1$ is 'almost' equal to $t_2$
$\dot{y}, \dot{z}$ are defined in a similar way.
A quantity of particular importance in physics is momentum. It is defined as
$m\bold{v} = (m \times x, m \times y, m\times z)$. From now on, for convenience, the symbol $\times$ will be omitted. You should know enough by now when we mean multiplication because of the context! If not, don't worry. Not everyone can be a barbarian, someone has to be pillaged!
The great discovery of early barbarians was that if you have another body, and it hits another body, if you add the two momentums before and after the collision, it does not change.
Already, you can see the applications such a discovery has on axe fighting!
Keep at it, wimps!
Every barbarian must have an acute understanding of physics, be it while fighting in glorious hand to hand combat, sailing your longship to ripe pillaging fields or calculating the trajectory a system undergoes through state space during a scattering process whilst remaining Lorentz -covariant.
Without being marred with philosophy, I shall make a few statements which are self evident (AGAIN, LOGIC!) and then construct the foundations of what may be called "Newtonian Mechanics", after one of the great barbarians: Newton the Madaxe (although most sources erroneously call him Isaac Newton)
Space can be considered as an ordered collection of numbers $(x,y,z)$ such that $x,y,z \in \mathbb{R}$. Upon fixing the 3 numbers representing some specific location in space, such as yourself, all other points may be unambiguously defined.
A simple way to do this systematically is to decide on a left and right, forward and back, up and down while the numbers $x,y$ and $z$ would correspond to a distance from the initial fixed point along those directions.
We can now know where everything is! However, bitter experience has shown us that things do not necessarily remain where they are right now. So therefore, we require a measure of time, $t \in \mathbb{R}$.
We also require another quantity mass, $m \in \mathbb{R}$, which encompasses "how much" of something there is.
From these we can extract the necessary physics.
An object of mass $m$ can therefore be labelled by a succession of different positions $(x,y,z)$ for every time $t$, with time constantly increasing.
Define the velocity $\bold{v} = (\dot{x}, \dot{y}, \dot{z})$ of an object by the change of positions divided by the change in times, as the change in time goes smaller and smaller.
In other words:
$\dot{x} = \frac{x_1 - x_2}{t_1 - t_2}$ where $x_1$ is the value of $x$ at time $t_1$, and so on, while $t_1$ is 'almost' equal to $t_2$
$\dot{y}, \dot{z}$ are defined in a similar way.
A quantity of particular importance in physics is momentum. It is defined as
$m\bold{v} = (m \times x, m \times y, m\times z)$. From now on, for convenience, the symbol $\times$ will be omitted. You should know enough by now when we mean multiplication because of the context! If not, don't worry. Not everyone can be a barbarian, someone has to be pillaged!
The great discovery of early barbarians was that if you have another body, and it hits another body, if you add the two momentums before and after the collision, it does not change.
Already, you can see the applications such a discovery has on axe fighting!
Keep at it, wimps!
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