∑ F = ma
… is a differential equation:
where acceleration a(t), velocity v(t), and displacement s(t) are all vectors and functions of time. This equation is second-order in position because the highest derivative is the second time derivative of position. Combined with the right boundary conditions, s(t) (also called the trajectory: path through space and time) can be determined.
This differential equation can be solved one component, or dimension, at a time. Let us focus on one of these, and call it the x component. The equations for y and z can be found exactly the same way.
If the graph of a(t) signifying acceleration in the x direction is constant
then the graph of v(t), the velocity in the x direction, is a straight line with slope a0
and the graph of x(t), the position along the x axis, is a parabola
It is also possible for the acceleration, or either of the initial velocity or initial position, to be negative. Thus the displacement/projectile motion formula is derived.
Sizes of Infinity
Everyone knows about infinity, but most people don’t know that there are actually different sizes of infinity: the transfinite numbers. This may out odd at first; after all, how do you define the size of something that is infinite?
For example: there are an infinite number of even numbers. But, to be more precise, there is a “countable infinity” of even numbers; in other words, you can enumerate them.
Say we start at 0 and call it the first even number; then we say that 2 is the second even number, -2 is the third, 4 is the fourth, -4 is the fifth, et cetera. In this way we can assign an integer to each even number. (Note that we can choose any numbering scheme we like; what’s important isn’t how we enumerate them, but rather that we can.) Therefore, the size of the even numbers is as big as the size of the integers, and that size is “countable infinity”, often represented by the symbol aleph zero. A similar argument can be given for odd numbers, positive numbers, negative numbers, primes, and so forth.
On the other hand: when one tries to count the real numbers, a problem arises. For any two real numbers there will always exist another between them. It is not possible to create a one-to-one correspondence between the real numbers and the integers; in other words, they are impossible to count. For a rigorous proof of this fact see Cantor’s diagonal argument.
Notice that between any two rational numbers (numbers of the form a/b) there is also always another number; nevertheless, we can construct a an isomorphism between the rationals and the integers. So, in fact, there are just as many rationals as there are integers! One such isomorphism is the Calkin-Wilf tree.