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 ( K) of the ball from release to just before hitting the ground as measured in frames S and S' separately, which one of the following is correct?
C. K is larger in frame S'.
In frame S (Earth's frame): Initial velocity of the ball, v_ i = 0 Final velocity of the ball just before hitting the ground, v_ f = 2gh (downwards) Change in kinetic energy in frame S is: K_ S = 1 2 m v_ f ^2 - 1 2 m v_ i ^2 = 1 2 m (2gh) - 0 = mgh In frame S' (moving upwards with constant speed u): Initial velocity of the ball, v'_ i = u = v (downwards) Final velocity of the ball, v'_ f = 2gh + v (downwards) Change in kinetic energy in frame S' is: K_ S' = 1 2 m (v'_ f )^2 - 1 2 m (v'_ i )^2 K_ S' = 1 2 m ( 2gh + v)^2 - 1 2 m v^2 K_ S' = 1 2 m (2gh + v^2 + 2v 2gh ) - 1 2 m v^2 K_ S' = mgh + mv 2gh Comparing the two expressions, we get K_ S' = K_ S + mv 2gh . Since m, v, g, h are all positive quantities, K_ S' > K_ S . Therefore, the change in kinetic energy is larger in frame S'. Answer: K is larger in frame S'.
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