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I was wondering if it is possible to calculate the mass of an object (what is commonly referred to as weight) if you know how long it takes to move from point A to point B (on earth). For example, if I take a barbell and put weights on it and lift it from the floor to above my head, is there a way I can use a calculation to confirm how much weight I just lifted. I know that mass = force/ acceleration but what if I don't know the force? Since I am lifting upward can gravity be used somehow to determine the force since gravity is a constant (well sort of)? Hopefully this makes sense 
Yes it is but only under certain conditions. If you just drop it and watch it accelerate you cannot calculate its mass because neglecting air resistance everything just accelerates in the same way. however if you had a known mass and a pulley system that was reasonably low loss connected the two together and then watched them accelerate pulling the lighter one upwards and the heavier one downwards (it does not matter which is the heavier) you can calculate the mass of the unknown object.
That does not achieve the objective of the question. The questioner was asking about a dynamic method of measuring the mass of an object and not a static one like a balance.