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Optimal gear ratio

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Hi, I am doing a project that involves an electric motor that winches an object upward pulling it by a string, over a given distance and I need help to calculate an optimal gear ratio, if that is possible. This is how far I have got...

Variables
torque/rpm
gear ratio

Constants
distance object travels
mass of object
motor input

to make it simple i am only using one gear, the motor shaft

acceleration of object = (f/m - 9.8)
velocity of string = RPM x pie x motor shaft diameter

assuming initial v=0,
the derivative of (f/m - 9.8)=RPM x Pie x Shaft diameter

i am trying to find a correlation between the average velocity the object takes to cross a given distance and the shaft diameter (gear ratio), im guessing its a parabola, but how can i do this? plz help thnx
 

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