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If work is a product of force and displacement, and force increases as acceleration increases, does these mean that work is dependent on acceleration? For instance, if I lift a block faster, does this mean that I did a greater work?

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It depends on the final state. If you lift the block to a height $h$ and let it go, the total work done will m $mgh+1/2\ mv^2$ where $v$ is the final velocity you were lifting the block. However, if you lift the block to a height $h$ and stop it, the total work done will be $mgh$ regardless of how fast you got there, since stopping the block will do negative work.

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force increases as acceleration increases

Don't you think that this should be the other way around. Acceleration increases as Force increases.

does these mean that work is dependent on acceleration?

If you want to interpret it this way then Work is dependent both on mass and acceleration which nothing but Force.If you consider mass to be same,again acceleration increases as force increases and not the other way around, though it does not make any significant difference in my view.

if I lift a block faster, does this mean that I did a greater work?

If you consider the same block,then to lift it faster you need to apply a greater force.If masses are different,then for different accelerations they may do the same work for a given displacement provided the product of mass and acceleration remains same.

Does work increases as accelaration increases?

So in nutshell,for a given displacement ,the answer is yes if you are considering same mass and no if you are considering different masses

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