If the jumping frog can give itself the same initial speed, regardless its direction. how is the maximum vertical height related to its maximum horizontal range?

A projectile motion is characterized by a motion whose path is alike to that of an arc. It already has its derived equations. For the maximum height H and the maximum range R, the equations are:

H = v₀²sin²θ/2g       ;       R = v₀²sin(2θ)/g

As you can observe, they are almost alike. Since they both contain v₀²/g, let's isolate this term for both equations:

v₀²/g = 2H/sin²θ
v₀²/g = R/sin (2θ)

Equating both equations:
2H/sin²θ = R/sin (2θ)
H = Rsin²θ/2sin(2θ)

This is the equation that relates H to R.

You get 6 for mechanical