I am working on a problem involving a car following another car. The leading car decelerates at a rate of 3 mi/s^2 until it comes to a complete stop. To save time from writing everything, I just included the variables down below.

The given 2nd order ODE is:

(x_fc - x_lc)* (d^2x_fc/dt^2) k* (dx_fc/dt) - k*v_lc = 0

dx_fc/dt = v_fc = v_fco

x_fc(t=0) = x_fco

​

# leading car, lc

# following car, fc

# initial speed, mph

v_lc = 30

v_fc = 30

​

# deceleration of lc

da_lc = -3 #mi/s^2

v_lc = 0

​

# sensitivity constant, k (mph)

k = 17

​

# length of car, l (ft)

l = 15

​

# distance from front of leading to front of following, d (ft)

d = 500

# average length of car, x (ft)

x_lc = 15

x_fc = 15

​

def dxfc_dt(t,y):

x_fc = y[0]

v_fc = y[1]

return [v_fc,k*(v_lc - v_fc)/(x_fc - x_lc)]

time = np.linspace(0,500,100)

z2 = solve_ivp(dxfc_dt,[0,0],time)

​

I believe I am on the right track but am confused with how to include some of the variables into the code. How do I include the distance between the cars, their average length, and the lead car's deceleration rate?