Question

Two balls are dropped, one from height H and the other from a height of 4H. How far apart should they be dropped (time wise) to hit the ground at the same time.

Answer 1

h=1/2 g t^2 is the basic formula.

H=1/2 g (t-to)^2 where to is the time the lower one is dropped, after the higher.
4H=1/2 g t^2
So, both balls hit at time t.

t=sqrt 8H/g
H=1/2 g (sqrt(8H/g)-to)^2
2H=g(8H/g-2to*sqrt(8H/g)+to^2)
so you have a quadratic...

to^2-2to* sqrt(8H/g)-2H/g

a=1 b=-2sqrt(8H/g) c= -2H/g

solve by the quadratic equation.