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While studying energy on Sears & Zemansky's University Physics, I came up with a doubt on the meaning of kinetic energy. The book gives two possible physical interpretations of this quantity.

So the kinetic energy of a particle is equal to the total work that was done to accelerate it from rest to its present speed [...] The kinetic energy of a particle is equal to the total work that particle can do in the process of being brought to rest.

I'm okay with the first meaning of KE but I don't understand completely the second one. How can the particle do work?

If we take a ball with velocity $v$ that meets a spring, the spring is compressed and the ball is stopped. But here it's the spring that does work on the ball, or vice-versa? The ball gains potential energy ($W_\textrm{spring}=-\Delta U_\textrm{ball}$) and mechanical energy is conserved in the process. Nevertheless I do not see if and where the ball does work here.

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    $\begingroup$ Both the spring and the ball do work. It's not one or the other. The force the ball applies to the spring does positive work on the spring and compresses it, while the force of the spring on the ball does negative work on the ball and slows it down. The two amounts of work cancel and the system energy stays constant. $\endgroup$
    – user55515
    Commented Apr 3, 2016 at 23:37

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I would say the system gains the potential energy in that example. The ball does work on the spring equal to the ball's initial kinetic energy.

Consider the direction of the force and displacement of each. The force the ball applies on the spring and the balls displacement as it applies it are in the the same direction so the work done is positive - it transfers energy to the spring & ball system. The spring's displacement and the force it applies on the ball are in opposite directions so the work it does on the ball is negative - the ball loses energy.

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    $\begingroup$ Objects don't have potential energy. Systems have potential energy. And when work is done on the ball by a spring, the ball does work on the spring as well, since forces come in action reaction pairs. It can seem mysterious when you are used to pretending the mass of one of the objects is infinite or zero when it is merely a large mass or a small mass, or pretending things do not move, that really do move and so on. When the potential energy of the system changes, this changes the kinetic energy of the system, and the parts of the system feel equal and opposite forces. $\endgroup$
    – Timaeus
    Commented Apr 4, 2016 at 1:21
  • $\begingroup$ @Timaeus Help me understand here. Is it inaccurate to say that spring gains potential energy? Should I say the spring & ball system gains potential energy? This does seem mysterious - that we can't disregard the ball once it has finished compressing the spring and just say the spring alone has the potential energy. $\endgroup$
    – M. Enns
    Commented Apr 4, 2016 at 1:40
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    $\begingroup$ Arguing about which object has the potential energy is not even a discussion to have. For something like gravitational potential energy, the formula is more obvious since it is a function of the distance between the parts of the system. But trying to have objects own the potential enegy and hand it all out is a bad bad idea that can lead to problems. For instance imagine a spring with two free ends with two balls rushing towards it. As it compresses the balls slow down and stop, then when the balls start to speed up. You wouldn't want it to give 100% of the potential energy to each ball. $\endgroup$
    – Timaeus
    Commented Apr 4, 2016 at 1:45
  • $\begingroup$ After all, that would double the non rest energy of the system. It is attaching a tiny mass spring to a huge mass earth that allows the simplifications people like to use. But they aren't the fundamentals of what is going on. And the spring can't keep the potential energy without a latch or something to keep it compressed, so the dynamics are determined by more than the spring length all by itself. $\endgroup$
    – Timaeus
    Commented Apr 4, 2016 at 1:50
  • $\begingroup$ OK, I see your point. I'll correct my answer. $\endgroup$
    – M. Enns
    Commented Apr 4, 2016 at 2:08
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"How can the particle do work?" Unless the particle was applying a force on a mass through some distance, the particle doesn't do any work. Work is done by a force through some displacement in the direction of the force.

Work is done in bringing that particle to rest by some acceleration through some displacement, sure. The wording is poor, but the meaning is essentially the same.

In your example, the spring does the work on the ball (Which is negative, btw), because the spring exerts a force on the ball through the displacement.

edit: The poster above me is more correct actually. You can describe it both ways because of Newton's 3rd Law. KE is the amount of work done to bring an object to rest, or the amount of work the particle can do before coming to rest.

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