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# Introduce Euler's Method

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## Richard Datwyler

on 6 May 2015

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#### Transcript of Introduce Euler's Method

Euler's Method

Tutorial
Y(t)
X(t)
y

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Spring !!
Mathematica
Excel
Recall the rocket problem
We made 3 or 4 positions x0, x1, x2, x3

and from those we made equations
x1 = x0 +v0*(t1-t0) +.5 a(t1-t0)^2
v1=v0+a(t1-t0)
x2 = x1+v1*(t2-t1) +.5(-9.8) (t2-t1)^2
If we had made this delta t constant, it would have followed this "Eulers" method technique.
In our rocket problem, what if we had added air resistance?
Recall we had to break the problem down into separate places where the acceleration was constant, and then solve the next one.
With air resistance our acceleration is not constant. We would need it for each small step we considered
The sum of all forces = mass*acceleration
F=ma
Jumping a bit ahead:
Our two forces are Gravity and Drag
so
F + F = ma
G
D
Trust me on this, and we'll learn it later but:
F = mg
G
and
F = .5 C p A v
2
D
m is mass
g is 9.8
C is aerodynamic coefficient
rho is density
A is area
v is velocity
This would make acceleration
or
Now using Eulers method we can use our equations of motion, and take this idea:

x1 = x0 +v0*(t1-t0) +.5 a(t1-t0)^2
v1=v0+a(t1-t0)
x2 = x1+v1*(t2-t1) +.5(-9.8) (t2-t1)^2
and write it with indexes that changes such as:
now the acceleration
Notice the signs on the two forces gravity and drag.

Is that always true?
UP
DOWN
Now let us see this in Mathematica
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