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So now we have equations that govern the
rates of change of the position and the velocity.
The rate of change of the position is the velocity,
and the rate of change of the velocity is the acceleration,
which is by Newton's law, force divided by mass,
and in a typical setting, we also know the position
at time 0 and the velocity at time 0.
Now we are going to use the computer to learn
what these equations are leading to, and the easiest
way for doing so is called the Forward Euler Method.
The most complex thing about that method may be
the name of Euler.
Euler was born in Basel, Switzerland, that is.
Most people these days use the German pronunciation
of his name -- Euler -- even though the
Swiss pronunciation may be different from that,
and in the U.S., you may also hear the pronunciation Euler.
I try to stick to the pronunciation Euler,
but forgive me if I, from time to time, fall back into saying Euler
Euler's idea was to solve these equations
by walking in small time steps.
If we start at the initial position--
x(0) and the initial velocity -- v(0),
what would happen after a short timespan we call h?
The position would approximately increase by h times the velocity.
If the velocity is 2 meters per second,
and we wait for 3 seconds,
we will be changing the position by 6 meters, for instance.
But, of course, we're using a much smaller time step.
Similarly for the velocity, after some small time h,
the velocity will be its original value plus
the time step times the acceleration, which is F/m.
So these equations will take us from
time 0 to time h approximately.
I'm cheating a little when I write equal signs here.
In the same fashion, we can get from time h to
time 2h doing another step.
This is what this second step looks like.
We know the position that we have reached at the end of the first step,
and we continue with the new velocity,
and that results in the new position --
similarly for the velocity.
You iterate this process over and over again
and find estimates for positions and velocities as you go.