Just read somewhere that the shortest path between two locations is an arc and not a straight line! eli5 how an arc is the shortest path for planes.
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Maybe not exactly what you were asking, but shortest can have 2 different uses. The shortest distance has been elegantly covered in other answered. There’s also the shortest time. For that, our hero is typically the “brachistochrone curve”. But that already assumes some things. Namely, the object is in a force field which tries to continuously and uniformly accelerates it downward, and the object is not propelled by anything else. In the real world, there is friction and wind resistance. However, a straight ramp will still lose to some sort of curve, it’s just not as simple the exact geometry of the curve (though arguably, the actual roots used to make the word “brachistochrone” imply that it is the shortest time path regardless of circumstances. In that case, for an object travelling at a constant speed, the brachistochrone curve is a straight line).
The main thing to consider is the worst case scenario. The path has a very very slight angle until just before it gets to point B, and then curves sharply downward. The ball or cart or whatever is making the journey will creep along ever so slowly for the vast majority of the journey, and will accelerate rapidly at the end. A straight line path will let the object accelerate uniformly through the entire journey. But even better is a curve that falls more steeply at the start, then flattens out a bit as it approaches the target. This way, it accelerates rapidly at the start, having a higher speed as it travels the flatter section. This higher speed makes up for the slightly longer path… To an extent. If you deform the curve too much, suddenly the extra length is not worth the extra speed.
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