Euler's Method with changing acceleration

Question

The goal is basically to create a plot of both altitude and velocity vs time . Given equations are: d^2y/dt^2=-g-b*v P(y)=P(0)*e^(-ky)

This is that I've attempted so far:

k = 1.256*10^-4;
b = 0.3266;
g = 9.8;
e = 2.71828;
inc = 0.1;
t[0] = 0;
v[0] = 0;
y[0] = 0;
P[0] = 17696;
P[y] = P[0]*e^(-k*y[t]);
y[t] = -g*y[t]*-b*y[t]*0.5*y^2;
v[t] = -g*v[t]*-b*v[t]*y[t];
a[t] = -g*-b*v[t];

i = 0;

While[
  y[i] > 0,
  P[i + 1] = P[0]*e^(-k*y[i]);
  t[i + 1] = t[i] + inc;
  y[i + 1] = v[i]*inc + y[i];
  v[i + 1] = a[i]*inc + v[i];
  a[i + 1] = -g*-b*v[i];
  i++
  ];
ptsvt = Table[{t[j], v[j]}, {j, 0, i}];
ptspt = Table[{t[j], P[j]}, {j, 0, i}];
ListLinePlot[ptsvt]
ListLinePlot[ptspt]

I'm pretty sure the main mistake is in the definition of y[t] and v[t]. I've tried several different equations and nothing seems to work. I'm not even sure if defining y[t] and v[t] is the right way to do it but I can't think of any other way. Any help would be greatly appreciated!

Answer 1

I do not think you intend to declare y[t] implicitly:

y[t] = -g*y[t]*-b*y[t]*0.5*y^2;

I think this should rather read:

y[t_] = -g*y[t]*-b*y[t]*0.5*y^2;

Then by specifying:

y[0] = 0;

and in the while loop:

y[i + 1] = v[i]*inc + y[i];

y would never change. So I change this to y[0]=1; This is certainly not right together with P[0] = 17696; but at least it gets you going. Change the other initial conditions according to your problem. These changes together with the correction from Henrik (a[0] = -g - b*v[0];):

Clear[t, y, v, P, a];
k = 1.256*10^-4;
b = 0.3266;
g = 9.8;
e = 2.71828;
inc = 0.1;
t[0] = 0;
v[0] = 0;
y[0] = 1;
P[0] = 17696;
a[0] = -g - b*v[0];
P[y_] := P[0]*e^(-k*y[t]);
y[t_] := -g*y[t]*-b*y[t]*0.5*y^2;
v[t_] := -g*v[t]*-b*v[t]*y[t];
a[t_] := -g*-b*v[t];

i = 0; inc = .01;

While[y[i] > 0,
  P[i + 1] = P[0]*e^(-k*y[i]);
  t[i + 1] = t[i] + inc;
  y[i + 1] = v[i]*inc + y[i];
  v[i + 1] = a[i]*inc + v[i];
  a[i + 1] = -g*-b*v[i];
  i++];
ptsvt = Table[{t[j], v[j]}, {j, 0, i}];
ptspt = Table[{t[j], P[j]}, {j, 0, i}];
ListLinePlot[ptsvt]
ListLinePlot[ptspt]

Now you can try to adapt the initial conditions or if you do not succeed, explain your problem.