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One would assume that:
$$t \propto h^\alpha m^\beta g^\gamma$$
Where t = time taken for object to fall, h = height dropped from, m = mass, g = acceleration due to gravity.
By doing some dimensional analysis one can find that:
$$t \propto h^\frac{1}{2} g^\frac{-1}{2}$$ and that t is independent of the objects mass.
From this, one can derive that:
$$t = C \surd\frac{h}{g}$$
Where C is some unknown constant of proportionality.
MY QUESTION:
How does one get from $$t \propto h^\frac{1}{2} g^\frac{-1}{2}$$ to $$t = C \surd\frac{h}{g}$$. I need to know all the mathematical processes and each step in detail.
$$t \propto h^\alpha m^\beta g^\gamma$$
Where t = time taken for object to fall, h = height dropped from, m = mass, g = acceleration due to gravity.
By doing some dimensional analysis one can find that:
$$t \propto h^\frac{1}{2} g^\frac{-1}{2}$$ and that t is independent of the objects mass.
From this, one can derive that:
$$t = C \surd\frac{h}{g}$$
Where C is some unknown constant of proportionality.
MY QUESTION:
How does one get from $$t \propto h^\frac{1}{2} g^\frac{-1}{2}$$ to $$t = C \surd\frac{h}{g}$$. I need to know all the mathematical processes and each step in detail.