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I was thinking about mechanics and energy and stumped myself with this scenario I made up:

Suppose you are in a room and you are motionless. In the same room, a 1 Kg ball is moving to your right with a speed of 1 m/s. You arrest the ball's motion, so the ball loses its one joule of kinetic energy and also becomes motionless.

However, I lied. The room is actually in space, and is going to my left at a speed of 100 m/s, and the ball is also going to the left from my perspective, but slightly slower at 99 m/s. From my perspective, the ball gains kinetic energy, despite the opposite being true from your perspective.

How is this possible?

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    $\begingroup$ You do work on the ball regardless of whether you slow it down or speed it up. I don't understand the question. $\endgroup$ – ACuriousMind Oct 31 '15 at 21:31
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    $\begingroup$ As 'CuriousOne' said, the value of a kinetic energy depends on the observer or frame of reference. You can't just say that the kinetic energy of a ball or other object is some value. One way or another you have to state what frame of reference you are using. Sometimes the frame of reference is obvious, so it isn't explicitly stated but that doesn't mean that it's not important. Your example merely illustrated the fact that kinetic energies do depend on the frame of reference. $\endgroup$ – Samuel Weir Oct 31 '15 at 23:42
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Try arresting the balls motion while standing on a path of ice.

What happens is the ball and that floor come to a relative rest. And any force exerted on the ball to slow it down or speed it up has an equal and opposite force exerted on the thing accelerating it.

You can analyze it in any frame and get a correct description (though kinetic energy depends on frame). You'll notice in the frame where the room was moving that the room slows down and hence loses some kinetic energy. Same in the center of momentum frame and the center of mass frame. In the frame where the room was initially at rest, the room actually gains some kinetic energy.

In every frame there is some relative motion. So there is some kinetic energy. That kinetic energy can get shared. And indeed, after the interaction the kinetic energy of each object (room and ball) has changed, but the total energy is conserved (though for a realistic situation with friction some energy has turned into heat and isn't readily available for work any more).

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This is really just a simplified version of Timaeus' answer, so please accept his answer not mine.

Anyhow, you're quite correct that the ball gains energy, but that energy doens't appear from nowhere. in any (inertial) frame total energy is always conserved. What you are seeing is some of the kinetic energy of me and the room being transferred to the ball. That is, when I catch the ball you see the ball speed up slightly and gain energy but you see me and the room slow down slightly and lose energy.

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