from numpy import array,arange

g = 9.81         # Acceleration due to gravity
a = 0.0          # Initial time
b = 10.0         # Final time
N = 1000         # Number of Runge-Kutta steps
h = (b-a)/N      # Size of Runge-Kutta steps
target = 1e-10   # Target accuracy for binary search

# Function for Runge-Kutta calculation
def f(r):
    x = r[0]
    y = r[1]
    fx = y
    fy = -g
    return array([fx,fy],float)

# Function to solve the equation and calculate the final height
def height(v):
    r = array([0.0,v],float)
    for t in arange(a,b,h):
        k1 = h*f(r)
        k2 = h*f(r+0.5*k1)
        k3 = h*f(r+0.5*k2)
        k4 = h*f(r+k3)
        r += (k1+2*k2+2*k3+k4)/6
    return r[0]

# Main program performs a binary search
v1 = 0.01
v2 = 1000.0
h1 = height(v1)
h2 = height(v2)

while abs(h2-h1)>target:
    vp = (v1+v2)/2
    hp = height(vp)
    if h1*hp>0:
        v1 = vp
        h1 = hp
    else:
        v2 = vp
        h2 = hp

v = (v1+v2)/2
print("The required initial velocity is",v,"m/s")