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I am going to explain this question through an example.

Suppose I lift an object I apply a force $mg $ then I apply additional force, that would be $ma $ so total force would be $m(g+a)$. My doubt is that the work done by a person in lifting a box by applying more force than the weight that should be $m(g+a) *h $ right?

I would love to know if my answer is right or if someone could kindly correct me

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That is correct. Also, since the gravitational PE increases by $mgh$ it is clear that the kinetic energy increases by $mah$

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Normally in lifting, you apply a momentary extra force to get the object moving, and then proceed at a constant speed. As Dale points out, the extra work goes into kinetic energy.

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