I'm basically trying to solve the opposite of the brachistochrone problem. You know how the brachistochrone is the shortest path between two points A and B, falling under the influence of gravity? Well I'd essentially like to find the shortest path between two points A and B, however I'd like the object to be travelling upwards (like a projectile).
Essentially, I'm trying to find the curve that minimises the time that a penalty kick in soccer takes between two distinct points. Here is a picture:
I know that the integrand of the T integral is $ds/v$. I'm just not sure how to pose this as a calculus of variations problem. Any help would be much appreciated.