Day 34
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(Concept Question)
Adding Air Drag
Review Euler Process
Trajectory with Drag
Euler Convergence
(Exercises)
ME123 Computer Applications I
Adding Air Drag
• Yesterday we used Euler’s method to solve
equations for which we knew the exact
solution
• We only did that for practice
• Today we will solve equations which have no
exact solution
ME123 Computer Applications I
Adding Air Drag
Suppose we have a ball falling:
𝑘𝑉 2
𝑚
𝑚𝑔
𝑑𝑉
= −𝑚𝑔 + 𝑘𝑉 2
𝑑𝑡
𝑑𝑉
𝑘
= −𝑔 + 𝑉 2
𝑑𝑡
𝑚
This one is much harder to solve analytically.
ME123 Computer Applications I
Adding Air Drag
This is the
only new term!
𝑘𝑉 2
𝑚𝑔
𝑑𝑉
𝑘
= −𝑔 + 𝑉 2
𝑑𝑡
𝑚
𝑑𝑦
=𝑉
𝑑𝑡
𝑉 0 =0
𝑦 0 =0
You will work with these equations in the exercises.
ME123 Computer Applications I
Review Euler Process
1. Replace differentials with small differences
𝑑𝑥
= 𝑓 𝑥, 𝑡
𝑑𝑡
∆𝑥
= 𝑓 𝑥, 𝑡
∆𝑡
𝑥𝑖+1 − 𝑥𝑖
= 𝑓(𝑥, 𝑡)
∆𝑡
ME123 Computer Applications I
Review Euler Process
2. Evaluate rhs at time 𝑖
𝑥𝑖+1 − 𝑥𝑖
= 𝑓(𝑥, 𝑡)
∆𝑡
𝑥𝑖+1 − 𝑥𝑖
= 𝑓(𝑥𝑖 , 𝑡𝑖 )
∆𝑡
ME123 Computer Applications I
Review Euler Process
3. Isolate 𝑥𝑖+1
𝑥𝑖+1 − 𝑥𝑖
= 𝑓(𝑥𝑖 , 𝑡𝑖 )
∆𝑡
𝑥𝑖+1 − 𝑥𝑖 = (∆𝑡) 𝑓(𝑥𝑖 , 𝑡𝑖 )
𝑥𝑖+1 = 𝑥𝑖 + (∆𝑡) 𝑓(𝑥𝑖 , 𝑡𝑖 )
ME123 Computer Applications I
Review Euler Process
4. March in time starting from initial condition
𝑥1 = 𝑥(0)
𝑥2 = 𝑥1 + (∆𝑡) 𝑓(𝑥1 , 𝑡1 )
𝑥3 = 𝑥2 + ∆𝑡 𝑓 𝑥2 , 𝑡2
𝑥𝑖+1 = 𝑥𝑖 + (∆𝑡) 𝑓(𝑥𝑖 , 𝑡𝑖 )
ME123 Computer Applications I
Trajectory with Drag
Launch a projectile with air drag:
𝑘𝑉
The air drag always acts to
oppose the motion of the
projectile.
2
𝑚𝑔
Its magnitude depends on the
square of the magnitude of the
velocity.
This one has NO exact solution.
It MUST be solved numerically.
ME123 Computer Applications I
Trajectory with Drag
Launch a projectile with air drag:
𝑚
𝑘𝑉 2
𝑚
y
𝑚𝑔
x
𝑑𝑉𝑥
𝑉𝑥 2
= −𝑘
𝑉
𝑑𝑡
𝑉
𝑉𝑥 0 = 𝑉𝑙𝑎𝑢𝑛𝑐ℎ cos 𝜃
𝑑𝑉𝑦
𝑉𝑦 2
= −𝑘
𝑉 − 𝑚𝑔
𝑑𝑡
𝑉
𝑉𝑦 0 = 𝑉𝑙𝑎𝑢𝑛𝑐ℎ sin 𝜃
𝑑𝑥
= 𝑉𝑥
𝑑𝑡
𝑥 0 =0
𝑑𝑦
= 𝑉𝑦
𝑑𝑡
𝑦 0 =0
You will work
with these
equations in
the Exercises.
ME123 Computer Applications I
Euler Convergence
• Euler gives you an approximate answer to
the equations
• The smaller ∆𝑡 is, the closer the answer is to
the correct solution
• When you don’t know the correct solution,
just keep making ∆𝑡 smaller until the answer
doesn’t change much anymore
ME123 Computer Applications I