When a reasonably fit person jumps for a basketball jump-ball situation or any jump from standing position with maximum effort placed they typically will not make more than 1m elevation in their center of mass. One can even argue that maybe 0.8m is more reasonable.
That would suggest the jump velocity from rest if standing on the edge is 1/2*mv^2=mgh or v=(2*g*h)^(1/2)=3.96m/sec
So lets say typically one can start from rest and assume they can accelerate during a bending and release of legs to full extension from rest without any prior running, just standing at the edge there (supported by some method to not drop prematurely) and they jump with immense effort delivering 4m/sec even 3.5m/sec for safety.
In that case the lowest part of the body say starts from height 4.27m and has to clear 3.35m plus room for error.
To leave some room for error ideally one wants to clear the point 3.35+0.3, -4.27 if one is jumping from what we view as origin at 0,0.
To achieve that you need to use the trajectory equation (that describes all the points x,y that this curve will ever pass from provided it doesnt hit any obstacles on the way to them)
https://en.wikipedia.org/wiki/Projectile_motion
where one has expressed y as function of x using as parameters the tangent of the angle of launch θ and the initial jump speed u.
y=Tan(θ)*x-1/2*g*(1+(Tan(θ))^2)/v^2*x^2
Your point of clearance is say x=3.35+0.3=3.65, y=-4.27 (since gravity is significant on earth the dominant velocity component at the edge of the pool is the vertical one for most possible by humans orbits , so i only need to worry about a horizontal clearance issue)
You need to find the set of u,θ that allow us to reach that point or better.
The above equation is a quadratic in tan(θ) and must have real solutions for u if reaching the x,y point is to happen. That will give you the condition that u,θ must satisfy.
As one can show easily using the discriminant of the tan(θ) quadratic (or see the above link under the section "Angle θ required to hit coordinate (x,y)")
u^4-g*(g*x^2+2y*u^2)>=0
this is the same as u^4-9.81*(9.81*3.65^2+2*(-4.27)*u^2)>=0
this has solution u>3.64m/2
But it gives very little room for error in the launch angle.
A safer speed like 4m/sec (but questionable whether achievable by most humans) would allow θ to range from 0 to 43 deg say that is doable as aim.
First of all i say immediately this is a no way reasonable jump because you can die or become paralyzed with some significant probability if you perform it and hit the floor or the edges.
To know if one can do it if you have to (without initial running speed possible due to the balcony geometry), just stand at the edge of the pool and jump with all your strength and try to launch with angle say 20-45 deg and so jump and see how far into the water you land. Take a video and examine the orbit.
The range is u^2/g*sin(2θ) so if for 20 deg you can land at least 1 m into the water then you have near 4m/sec. To be safe try to go for 1.2 1.3 m with a few test jumps. If you have that then maybe you can do it.
Air resistance will inhibit the actual risky jump a little but i doubt its significant at such low x component speeds. You will be in the air for about a second it seems and the x position remains close to the no air resistance solution.
Try this to see what is going on
https://www.desmos.com/calculator/reesidrq8h
To jump with such speed as near 4m/sec and such small angle you need to have some support in your launch segment that allows you to extend and accelerate in a direction that is far from vertical that may make it harder. Just do a test jump as i suggested from floor level and see how far you land. If you cant clear 1.1-1.2m its probably bad. Without any support in place if one stood up there in the balcony edge and jumped, there is such a significant risk something can go wrong, that its a very bad idea. At the very least if one dared to do it put multiple mattress type protective material (eg some of them used in gyms) all the way to the pool edge.
If you land bad you will at least not die but can still get injured even with the "mattress".
Ps:If you have to do it because your life depends on it i bet you can clear it.
Last edited by masque de Z; 07-09-2019 at 06:28 AM.