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1. The frame fixed to the earth's surface :-

Initially,

Potential energy of the object = [tex]mgh[/tex]

Kinetic energy of the object = [tex]0[/tex]

Finally,

Potential energy of the object = [tex]0[/tex]

Kinetic energy of the object = [tex]mgh[/tex]

2. A frame moving towards the earth's surface at a constant speed of [tex]\sqrt{2gh}[/tex] :-

Initially,

Potential energy of the object = [tex]mgh[/tex]

Kinetic energy of the object = [tex]\frac{1}{2} m(\sqrt{2gh})^2[/tex] = [tex]mgh[/tex]

Finally,

Potential energy of the object = [tex]0[/tex]

Kinetic energy of the object = [tex]0[/tex]

It appears that in the second case, the energy is not conserved. What's the flaw in the above reasoning?