Euler loop list

Question:

I want to plot a rocket trajectory with a loop using Euler’s method. My code is(you can skip this, the error is below it):

mo = 1000
m = 900
q = 2.8
u = 3700
vo = 0
xo = 0.0001
g = 9.81
t = np.arange(0,int(1001))
x0 = 0.0001
v0 = 0
N = 1000
at = 1/1000

Now velocity and height equations:

def v(t):
    return  u*np.log(mo/(mo-q*t))-g*t

def az(t):
    return xo + vo*t - 0.5*g*t**2 + u*t*np.log(mo) + (u/q)*((mo - q*t)*np.log(mo 
    - q*t) + q*t - mo*np.log(mo))

z = xo + vo*t - 0.5*g*t**2 + u*t*np.log(mo) + (u/q)*((mo - q*t)*np.log(mo 
- q*t) + q*t - mo*np.log(mo))

And here is the acceleration which I turned into a list because of the error:

aj = -g + (q/(mo-q*t))*u
ajj = np.array(aj).tolist()

So here is the loop where I am trying to solve it in an analytic way with Euler’s method:

t = np.zeros(N + 1)
t[0] = 0
z = np.zeros(N + 1)
z[0] = x0

v = np.zeros(N + 1)

v[0] = 0


for k in range(0,N):
    v[k + 1] = v[k] +ajj*z[k]*at
    z[k + 1] = z[k] + v[k + 1]*at

And it gives me this error:

ValueError                                Traceback (most recent call last)
<ipython-input-193-d1513ed52aa0> in <module>()
 11 
 12 for k in range(0,N):
 ---> 13     v[k + 1] = v[k] + ajj*z[k]*at
 14     z[k + 1] = z[k] + v[k + 1]*at
 15 

 ValueError: setting an array element with a sequence.

I know what it barely means. The error is because of the list ajj but I don’t know how to fix it in order to plot z along the time. Thanks for helping. Sorry for the long question I’m just starting programming.

Asked By: fghjk

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Answers:

Your ODE system is m'(t)=-q, az'(t)=v(t), v'(t)=u*q/m(t)-g, here with inserting m(t)=m0-q*t. It always helps to separate the system dynamics from the solver via an ODE function,

def ODEfunc(t,y): az, v = y; return np.array([ v, -g+q*u/(m0-q*t) ]);

so that you can then iterate the Euler steps as per cook book formula

t, y = t+h, y + h*ODEfunc(t,y)
Answered By: Lutz Lehmann
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