B Difference in energy not the same in different reference frames

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Consider a particle moving at velocity v1 in a straight line. If it were to receive a boost so that its velocity increases to a new higher velocity v2, the change in its kinetic energy as measured by a stationary observer would be 1/2 m (v2^2 - v1^2).

However, if we consider an observer moving at a velocity v1 next to the particle, the relative velocity of the particle changes from 0 initially to (v2 - v1), so its change in kinetic energy as measured by this observer is 1/2 m (v2 - v1)^2.
These quantities are evidently not equal unless v1 = v2, but surely the change in kinetic energy should be the same in different frames of reference? I was wondering what mistake or misconception I have made because this has been bugging me for a little while.

Thank you in advance!
 
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surely the change in kinetic energy should be the same in different frames of reference? I was wondering what mistake or misconception I have made
Your analysis is correct. You have not made a mistake. The change in kinetic energy is indeed different in different reference frames.

You may be concerned about conservation of energy. Conservation is a separate issue. KE is frame variant, but energy is still conserved
 
Your analysis is correct. You have not made a mistake. The change in kinetic energy is indeed different in different reference frames.

You may be concerned about conservation of energy. Conservation is a separate issue. KE is frame variant, but energy is still conserved
Thank you for your kind response, that makes a lot of sense.

The problem I was doing at the time concerned the minimum velocities and consequently energies required to get a satellite from the equator, moving at a linear speed of 0.5 km s-1, into a stable circular orbit of theoretical height 0 metres with a linear velocity of 7.9 km s-1, all relative to a stationary observer in space.

Suppose some amount of work E is done on the satellite by forces from the exhaust gases over some distance during "lift off" to increase its kinetic energy from that corresponding to 0.5kms-1 to that corresponding to 7.9 kms-1.

Switching to the perspective of someone moving with the satellite around the equator before "lift off", the exhaust gases also do some work on the satellite to increase its kinetic energy from 0 initially to that corresponding to 7.4kms-1 (the relative velocity from the moving observer).

The situations are identical, so it appears that the amount of work done by the rockets should also be the same. However, like you mentioned earlier, the differences in kinetic energy are different in different reference frames, so the amount of work we calculate as necessary to get the satellite to this speed are apparently different.

I have the feeling that the amounts of work needed to get the rocket to the required velocity are also different depending on the reference frame, but cannot find a concrete source of this. Furthermore, what value of energy would we quote if trying to figure out e.g. how much fuel to use?

Thank you so much for your help!
 
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jbriggs444

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The situations are identical, so it appears that the amount of work done by the rockets should also be the same.
Consider the work done by the rocket on the exhaust stream.
 
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I too actually posted a similar question regarding friction and energy loss. Yeah even I dont know how changing your frame of referece will affect the energy dissipation
 

jbriggs444

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I too actually posted a similar question regarding friction and energy loss. Yeah even I dont know how changing your frame of referece will affect the energy dissipation
If you count all the participating bodies, energy dissipation does not depend on the choice of reference frame. It is an invariant. It is only if one restricts attention to a single object (e.g. the rocket) that a net effect is seen.

This derives from Newton's third law. The same force pairs are there in both frames. The same points of application are there in both frames. The change of frame changes the relevant velocities of the participating objects equally. It follows that the works done by the two members of the force pair are changed equally and oppositely.
 
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Furthermore, what value of energy would we quote if trying to figure out e.g. how much fuel to use?
As @jbriggs444 alluded to in post 4, the key is to consider the KE of the exhaust. I would go so far as to say that all confusions about energy conservation in rockets is due to a failure to consider the energy of the exhaust.

If you work it out, it turns out that for a non-relativistic rocket the energy in the fuel is frame invariant and any differences in the rocket KE from frame to frame are entirely compensated by differences in the exhaust KE.
 

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