# Q & A: work in different frames

Q:
Hello. At start, please forgive me my ignorance if this question is stupid (probably it is).Small example: We have a m=1kg body, with resultant velocity vector = 0. We give to it v=1m/s, it is 0,5J.Now, let's increase its velocity by another 1m/s (so it has 2m/s now). We need to use not 0,5J, but additional 1,5J. If we want to increase its speed again and again, we need to use more energy everytime.And here is my question. We are on Earth. Absolute velocity of Earth and every object on it is (for example, i do not know exact value, but I am sure it is quite huge) approx. 100 000 meters per second - I can say that my sugar pack (1kg) has kinetic energy equal to 5 gigajoules (still, it is relatively immobile for me). But, if I throw it and give to it additional velocity v=1m/s, from my point of view it has that v=1m/s, but its absolute velocity is in fact 100001m/s (taking direction and sense as the same). By mine point-of-view i need 0,5J, but in fact, I need 100000,5J to do this action (not relatively, energy cost when increasing velocity from 100000m/s to 100001m/s for 1kg body).How is it working truly (I guess there is an answer in relativity theory)? Thanks for help :).
- Michael (age 17)
Poland
A:

That's a great question. Our answer will use relativity, but only needs simple old "Galilean" relativity, not the more subtle modern relativity. As you say, the amount of energy you give to the pack when you throw it comes out completely different in different frames. Yet as far as your sensation of using up energy goes there's  no difference in the different frames. The amount of chemical fuel you need to burn is just the same as in the frame where you, the Earth, and your pack were all initially at rest. Where can all the extra energy come from in the frame where you are all initially moving in the direction of the throw?

It comes from the initial kinetic energy of the Earth. Conservation of momentum says that the Earth must slow down a bit for that forward throw. It's not hard to check that the lost kinetic energy of the Earth exactly makes up the missing part you were worried about.

Incidentally, if you throw the pack the opposite direction, you reduce its kinetic energy, even though you still have to burn some chemicals to do it. Where does all that extra energy go? It goes into increasing the Earth's kinetic energy, by the same momentum conservation argument.

There's one false assumption in your question. There is no one frame that's "true". You can describe the facts in any of these frames using the same laws of physics.

Mike W.

(published on 11/23/2015)