Relationship between the fall time and projected area

I am investigating the relationship between the fall time of a paper tray and its projected area.

In this investigation I have been using the following variables:

• Controlled Variable: The air density, the height from which we drop the tray, the shape of the tray, its mass and the material it is made from (paper).
• Independent Variable: The cross-sectional area of the paper tray.
• Dependent Variable: The time it takes for the tray to fall on the ground

I found the relationship for when the paper tray is traveling at terminal velocity. This relationship I devised by using the formula for the speed of a object traveling at terminal velocity:

Terminal velocity can be considered uniform motion (acceleration = 0) and so the above formula can be rewritten in the following way:

But all this excludes the acceleration period before a object reaches terminal velocity. I suppose that because the paper tray is only 0.016 kg and the paper tray has a fairly high drag coefficient 1.31 that the acceleration period can be ignored since the tray reaches terminal velocity quite soon, but then again the trays are being dropped from only 2 meters.

Can I ignore the acceleration period? Is there a better equation for the relationship? Can you suggest further reading?

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Try videotaping it while you drop it. Play it back frame by frame, you should be able to tell how long it takes to reach terminal velocity by inspecting the playback. I'd suspect that if it's thin enough, it ought to reach terminal velocity quite quickly, but a videotaping could make sure. – DumpsterDoofus Apr 20 '14 at 23:37
Just make sure that when video taping it, you have clear height indicators in the field of view. This will help you measure speeds more easily. – Kyle Kanos Apr 21 '14 at 1:42