Evaluate the travel time of a particle over a broken line on the gravity field

Question

Given a list of 2-uples on python [(x_0,y_0),(x_1,y_1)...(x_n,y_n)], i need to evaluate the travel time of a particle placed at the point (x_0,y_0) with the initial velocity v_0 through the broken line formed by the point of the previous list.

exemple

By application of the second Newton's law, i came up with this algorithm, but i'm not confident about it :

def travel_time(v_0,l):

    n=len(l)
    t=0
    v=v_0
    a=0
    T=0
    g=9.8
    for k in range(n-1):
        a=np.arctan(abs((l[k][1]-l[k+1][1])/(l[k][0]-l[k+1][0])))
        t=(l[k+1][0]-l[k][0])/(v*np.cos(a)) 
        v=np.sqrt((np.cos(a)*v)**2+(-g*t+v*np.sin(a))**2)
        T += t 
    return T 

Can you help me to spot some potential mistakes, or submit improvements for the algorithm?

Answer 1

There are several things you should correct.

1) Your formula for the final velocity $v$ is wrong: you should use instead $$ v=\sqrt{v_0+2gL\sin\alpha}, \quad\text{where}\quad L={x_{k+1}-x_k\over\cos\alpha}\quad \text{is the length of $k$-th segment}. $$

(Side note: you'd better use $L=\sqrt{(x_{k+1}-x_k)^2+(y_{k+1}-y_k)^2}$ )

2) You compute travel time on $k$-th segment as $L/v_0$, but that is not accurate as velocity is not constant; you should use instead $$ t={2L\over v_0+v}, $$ where $v$ is the final velocity computed above.

Of course you should at the end of each cycle set $v_0=v$.