Kinetic Energy Gain: Why Is There a Difference?

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When a car moving at speed u throws a ball at speed v relative to itself, the kinetic energy gained is perceived differently by the car's occupant and an observer on the ground. The occupant measures a gain of mv²/2, while the ground observer calculates a change of m[v² + 2uv]/2, highlighting the discrepancy in energy perception due to different reference frames. This difference arises because the kinetic energy of both the car and Earth changes, affecting the total energy calculation. The discussion references a previous thread on "DDWFTTW" propulsion, emphasizing the importance of agreeing on a reference frame when discussing energy sources. The example of a roller skater throwing an object illustrates that the energy of the thrown object includes both the skater's effort and their momentum.
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Say a car is moving with speed u. The guy in the car chucks a ball with speed v relative to himself. The gain in KE,as recorded by him will be mv2/2. For an observer on the ground, i.e. non-moving frame, the change in kinetic energy is [(v+u)2 - u2]m/2 = m[v2 + 2uv]/2. Why is there a difference in energy gained? I am not able to put my finger on it.
 
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Because there is a difference in how much the kinetic energy of the Earth and car has changed. (Do the math, imagine what happens when a roller-skater throws a brick.)

There was a long thread here previously about "DDWFTTW" propulsion, and your topic is the reason why it never made much sense there to argue (without agreeing on a reference frame) whether the cart's power was "coming from" either the air at the propeller or the ground at the wheels.
 
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I'm going to use the example Cesiumfrog used.

Think of a roller skater throwing a ball or a brick, the amount of energy the object thrown has is not just the energy that the roller skater threw the object, but also the momentum energy the roller skater had while he threw it
 
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