TL;DR: Please point out any mistakes in solution . If there isnt why the work done by force come out to be equal to work done my gravity , and why when not moving it slowly there is an addittional term . If the work done is equal to Wd by gravity , I have read that work done by gravity always remain mgh if it has disp through height h , why extra kinetic erg . Why diff answers in both methods . Thank You

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For the second part, there is the work against gravity plus the work done to accelerate and give it a velocity. Whenever there is a velocity, there is kinetic energy.