IS SO LAST NOPIX
. -- ongomome
-- posted by SilentBot
Time for some analysis.
solve differential equations
The lines and arrows are not just a nice picture.
They actually show how this differential equation is solved.
So what do we see here?
Newton throws an apple diagonally
and then figures out its path through space and time.
We start with this:
x''(t) is the second derivative of x(t) - which is the function describing the apple's path or, more specifically, it's vertical position over time, and that is the function we are trying to find here.
g - the "standard gravity
and there is a minus sign because it's pointing downwards and we assumed that upwards is the positive direction and downwards the negative one.
-g is a constant value, so we get a horizontal line:
from x''(t) - (second derivative - also known as the rate of change of the rate of change of the vertical position, in other words vertical acceleration) we can go to x'(t), the first derivative by a process known as integration.
Integration can be, like this frame says, "extremely difficult
Luckily, here we have a simple example which can be solved without too much effort.
To go from x''(t) to x'(t) we have to treat each value of x''(t) as the rate of change of x'(t) in the same point of time.
Graphically, if we represent the values of x''(t) as arrows then we build x'(t) by stacking the arrows on top of each other:
So we can think of integration as arrow-accumulation
Ok, so this way we can get the next value of x'(t) basing on current value of x'(t) and x''(t) but how to get the first
the answer is - put there the original speed at which the apple was thrown. (v0
on the graph)
if we repeat the same thing once again we get from x'(t) (first derivative - also known as the rate of change of the vertical position, in other words vertical speed) to x(t) (the function which we are looking for):
And again we just have to put x0
as the initial position.
And here is our solution.
Of course, in reality we have to use a lot more points than the arrows shown here, much closer to each other than shown here,
otherwise we will get results which are too inaccurate, especially with functions which are not as simple as these shown here.
But that's what computers are good at.
That makes them a good tool for such things.
Also, what this frame does numerically (by accumulating values in lots of points) the previous frame does analytically (by actually finding the exact formula):
ETA: and now there is another frame, with gravity wells.
BE CAREFUL AND DON'G
T FALL INTO THE GRAVITY WELL