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I was wondering, from where does the opposite force and thus energy comes from when we apply force.

For example, lets say there are two persons (P1 and P2) on the universe, and no force is applied to them. P1 applies a force to P2. P2 will start to move. P1 lost some energy to do that. Now, according to the third law of newton, P1 will also get a force from P2 as a reaction.

But from who does the energy of reaction comes from? I mean, I found it hard to believe that P2 is the one who losses energy, just for the sake of reaction. So, from where does this force, and thus energy, comes from?

I am not an expert, I just know the basics. So please, be tolerant :P

Qmechanic
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Middem
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There aren't two separate forces, there is just one force. P1 feels a force being exerted by P2, and P2 feels a force being exerted by P1.

The work done by the force goes into kinetic energy, and this energy is shared between P1 and P2. To take your model of P1 and P2 floating unaffected by any other forces, when P1 pushes on P2 that will start P1 moving in one direction and P2 moving in the other direction so the work done goes into accelerating both people.

I'd guess that you're thinking because it's P1's arm doing the pushing the push must be in some way asymmetrical. But suppose you model P1's arm by a compressed spring between the two people. In this case it's obvious that the spring just pushes the two people apart. The fact that one end of the spring/arm is connected to P1 (and therefore stays attached to P1 when P1 and P2 separate) makes no difference.

John Rennie
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Force is not energy. Energy is defined as $\vec F\cdot\vec s$--in layman's terms "force times distance moved in the direction of the force". You do not need energy to create a force--you need energy to use a force to push something.

In fact, without reaction, energy would not be conserved.

Let's analyse your situation now that we know the relation between energy and force. Let's say $P_1$ pushes $P_2$ with a force of 1 N, by a distance of 1m forward. Work done by $P_1$ on $P_2$ is $1\:\mathrm{N}\times1\:\mathrm{m}=1 \:\mathrm{J}$. Work done is just another word for energy transferred, so there is one joule of energy transferred from $P_1$ to $P_2$.

Now, let's analyse the situation from $P_2$'s point of view. $P_2$ exerts a reaction force of $1 \:\mathrm{N}$ back on $P_1$, and the distance is the same, but they are in opposite directions. Since the formula is a vector dot product, the "opposite directions" makes the work done negative, i.e, $-1\:\mathrm{J}$. This means that $P_2$ transferred $-1\:\mathrm{J}$ to $P_1$ in the exchange, or, in other words, it recieved $1\:\mathrm{J}$. This is consistent with what we already know.

Now, let's look at the overall system. The net work done by the two forces in it is $1\:\mathrm{J}+(-1\:\mathrm{J})=0\:\mathrm{J}$, so there is no net energy gain or loss. Energy is conserved.

Manishearth
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