Disclaimer: I am a 16yr old high school kid and my mathematical communication skills are probably not up to scratch for this website.
A while ago I thought of this problem: (there may be some holes in my thinking here)
Jeremy the librarian has had enough and is about to jump off the top floor of his block of flats. Luckily the fire service has shown up just in time with their emergency slide building kit. All they have to do is enter a function into the display and it will be graphed into a slide for Jeremy to land on. Unfortunately, the peace of equipment can only build up to a maximum height.
Which function should they enter into the display to maximise Jeremy's chance of survival?
(Assume there is no acceleration due to gravity once Jeremy touches the slide. Constant deceleration along the x-axis is the best way for him to survive. Assume the slide is frictionless)
I got $x^{\frac{x}{\sqrt{1-x^2}}}$ as a possible answer though it wasn't very well founded.
I apologize if none of this makes any sense to you (it does to me but that's because I have been thinking about it a lot).
This isn't an answer to your question, but a similar question. You state that, when the person hits the slide, gravity ceases to exist, and the best case scenario is one where they experience constant acceleration in the horizontal direction. Although this might be an interesting problem to try and solve, and the 'no gravity' assumption can be considered as simplifying for the case where the force from the slide is much greater than that of gravity, this is clearly not the case, gravity will still exist, and the best case scenario is constant force from the slide.
This is precisely the problem that faces roller coaster designers when they want to build a loop, the 'perfect' loop is one where the roller coaster experiences constant force from the track as it goes round (though many loops will differ slightly from this to create different sensations, or for engineering reasons). You can see a discussion of loop design here. The perfect slide would look like the lower section of a loop, where it transitions from vertical to horizontal.