A ball is dropped from rest from height h above the ground in frame S (the Earth's frame). Another frame S' is moving upwards with constant speed u relative to the Earth. In frame S', the ball has initial downward speed v. Pertaining to the change in kinetic energy (AK) of the ball from release to just before hitting the ground as measured in frames Sand S' separately, which one of the following is correct?

In frame S (Earth), the ball starts from rest (vinitial = 0) and reaches a final speed vf = √(2gh). The change in kinetic energy is ΔKS = 1/2mvf2 = mgh.

In frame S', which moves upward with speed u relative to Earth, the ball has an initial downward speed v (where v = u). When the ball hits the ground, its speed relative to Earth is vf, so its speed in frame S' is (vf + v). The change in kinetic energy in S' is:
ΔKS' = 1/2m(vf + v)2 - 1/2mv2
ΔKS' = 1/2m(vf2 + 2vfv + v2) - 1/2mv2
ΔKS' = 1/2mvf2 + mvfv = mgh + mvfv.

Since m, vf, and v are positive, ΔKS' is greater than ΔKS. Thus, the change in kinetic energy is larger in frame S'.