def second(de, x, tstart, tend, steps):
    t=tstart 
    dt=float(tend-tstart)/steps 
    for i in range(steps):
#  step forward one step in x
       dx=de(t,x)
       xt=[a+b*dt for (a,b) in zip(x,dx)] 
       t+=dt
#    find new derivative
       dx2=de(t+dt,xt)
#  step forward using average if two derivatives
       x=[a+(b+c)*dt/2 for (a,b,c) in zip(x,dx,dx2)] #step forward again.
    return x

def ball(t,x):
  g=9.8
  v=x[0]
  return [g,v]



p=[0,0]
print second(ball,p,0.0,1.0,10)
print second(ball,p,0.0,2.0,10)
print second(ball,p,0.0,3.0,10)
print second(ball,p,0.0,4.0,10)
print second(ball,p,0.0,5.0,10)
p=[0,0]
p= second(ball,p,0.0,1.0,10)
print p
p = second(ball,p,1.0,2.0,10)
print p
p = second(ball,p,2.0,3.0,10)
print p
p = second(ball,p,4.0,5.0,10)
print p
p=[0,0]
p= second(ball,p,0.0,1.0,10)
print p
p= second(ball,p,0.0,1.0,10)
print  p
p= second(ball,p,0.0,1.0,10)
print p
p= second(ball,p,0.0,1.0,10)
print p
p= second(ball,p,0.0,1.0,10)
print p
p=[0,0]
print second(ball,p,0.0,5.0,1000)