python function return using numpy.array()

I am writing a python code for my Ordinary differential Equations class which I have to model Newton's 2nd Law of motion using a numerical method (RK4). I believe my code is correct as i have worked out the differential equations by hand myself and verified the results.

I would Like to cut down the length of my code my grouping the similar ODE functions together but I have come across an error regarding the return type. Below is my work so far:

import numpy as np
import matplotlib.pyplot as plt

t0 = 0.0                # Start time
tf = 10.0               # End time
h = 0.001               # step size
t = np.arange(t0,tf,h)  # Time Points
v0 = 20                 # Initial velocity in m/s
theta = 45              # Launch angle
Uy = v0 * np.sin(theta*np.pi/180) # Initial velocity in y component
Ux = v0 * np.cos(theta*np.pi/180) # Initial velocity in x component


def velocity_y(t,v):
    g  = 9.81               # Acceleration of free fall
    dVy_dt = -g
    
    return dVy_dt

def velocity_x(t,v):
    dVx_dt = 0
    
    return dVx_dt

def velocity_xy(t,v):
    g  = 9.81               # Acceleration of free fall
    dVy_dt = -g
    dVx_dt = 0
    
    return np.array([dVy_dt,dVx_dt])

p1,p2 = velocity_xy(t,v)

def rk4( f, x0, t):        
    n = len( t )
    x = np.zeros(n)
    x[0] = x0
    for i in range( n - 1 ):
        h = t[i+1] - t[i]
        k1 = h * f( x[i], t[i] )
        k2 = h * f( x[i] + 0.5 * k1, t[i] + 0.5 * h )
        k3 = h * f( x[i] + 0.5 * k2, t[i] + 0.5 * h )
        k4 = h * f( x[i] + k3, t[i+1] )
        x[i+1] = x[i] + ( k1 + 2.0 * ( k2 + k3 ) + k4 ) / 6.0

    return x

r = rk4(velocity_y,Uy,t)
r1 = rk4(velocity_x,Ux,t)
plt.plot(t,r1,'gx',linestyle='-',label='Velocity in y direction')
plt.xlabel('t or x')
plt.ylabel('v(t) or y(x)')
plt.legend()
plt.grid()
plt.show()

I implemented my ODE functions as below:

def velocity_y(t,v):
    g  = 9.81               # Acceleration of free fall
    dVy_dt = -g
    
    return dVy_dt

def velocity_x(t,v):
    dVx_dt = 0
    
    return dVx_dt

def velocity_xy(t,v):
    g  = 9.81               # Acceleration of free fall
    dVy_dt = -g
    dVx_dt = 0
    
    return np.array([dVy_dt,dVx_dt])
p1,p2 = velocity_xy(t,v)

I am trying to group similar ODEs into the same function kinds so as to avoid over creating new functions just to define new ODEs.

def velocity_xy(t,v):
        g  = 9.81               # Acceleration of free fall
        dVy_dt = -g
        dVx_dt = 0
        
        return np.array([dVy_dt,dVx_dt])

I believe the error is:

p1,p2 = velocity_xy(t,v)

but I do not know the proper syntax. The error from the program:

    NameError                                 Traceback (most recent call last)
<ipython-input-37-4ce3cb1658cb> in <module>
     30     return np.array([dVy_dt,dVx_dt])
     31 
---> 32 p1,p2 = velocity_xy(t,v)
     33 #-----------------------------------------------------------------------------
     34 

NameError: name 'v' is not defined

I would appreciate any helpful answers and comments.