Kinetic Energy Mystery

Suppose you have a spaceship powered by batteries. The first battery speeds up the spaceship from 0 to velocity = V. After the first battery completely drains, we switch to the second battery, which is exactly the same as the first. This battery should speed up the spaceship from velocity = V to velocity = 2 * V. We know that, because if we look from a distant planet flying at velocity = V in same direction, the second battery should speed up the spaceship from 0 to V relative to that planet.
Now, the kinetic energy gained with the first battery is (1/2)mV^2. The kinetic energy gained from the second battery is (1/2)m(2V)^2 - (1/2)mV^2 = (3/2)mV^2. So the second battery generates 3 times as much energy as the first one, even though they are exactly the same. Why?

#Mechanics

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Comments

Work is a relative quantity, so when you said that the battery did the same amount of work from V to 2V as from 0 to V, you are talking about one scenario in the frame of the planet and the other in the frame of space (or something).

this is my explanation, feel free to correct

Rohan Joshi - 1 month, 3 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$