Distance along a line at a given time, taking speed change into account

Question

I'm trying to calculate the distance along a line at a certain time, taking the speed change into account.

What I have, two points (lat, lng), the time between the two points, the distance between the two points and the speed at both points.

What I want, the distance (along the line) at a specific time, taking the initial speed and end speed into account.

I'm not really sure what formula I should be using for this (or if it's possible with the data I have), I've been googling, but not really getting anywhere.

Any help would be appreciated, please ask if you need any more information.

(NB. wasn't even sure what tag to use, please update if you think another tag makes more sense)

Thanks for taking the time to read this

Answer 1

Let's assume that the acceleration is constant. Then $accel=(v_1-v_0)/(t_1-t_0)$. Where $v_1$ is end speed and $v_0$ is starting speed and $(t_1-t_0)$ the time to do the whole journey. The distance along line at time $t$ is given by $d=v_0*(t-t_0)+ (1/2)*accel* (t-t_0)^2$.

If the acceleration is constant then if you put $t=t_1$ this should give you the distance between the points which you already know.