Is kinetic energy affected by gravity? For example, at low velocities (relative to c), the equation E=1/2mv^2 should be accurate enough here. So if I observe a 10 kg mass moving at 1x10^3 m/s in a given direction relative to me, I associate a specific value with its kinetic energy. Now this kinetic energy remains always with the mass rather than simply being dissipated throughout space (Newton's Laws). Velocity can increase or decrease but the energy is always with the mass. Why? What holds this energy in place and how can it be expressed mathematically?

Question

Is kinetic energy affected by gravity?  For example, at low velocities (relative to c), the equation E=1/2mv^2 should be accurate enough here.  So if I observe a 10 kg mass moving at 1x10^3 m/s in a given direction relative to me, I associate a specific value with its kinetic energy. Now this kinetic energy remains always with the mass rather than simply being dissipated throughout space (Newton's Laws).  Velocity can increase or decrease  but the energy is always with the mass.  Why?  What holds this energy in place and how can it be expressed mathematically?

turingtest · Answer

Energy is not just "in the mass" as you say, for instance it can exist in a relation between objects. A spring has potential energy due to its compression or extension. Planets have potential energy as a result of their configuration in space. Photons have no mass, but still have energy depending on the frequency of the wave.

I'm not sure what you mean by "holding the energy in place". In a swinging pendulum, for instance, energy is being transferred from kinetic to potential. I suppose you could say that the kinetic energy is "in the mass" in the sense that it is a result of the velocity that the mass has. However at the top of the swing (when the pendulum pauses for a moment) v=0 so the kinetic energy is transferred to potential. Where is this energy? Answer: nowhere. It is a result of the relation between the earth and the mass of the pendulum. It is certainly not just "in the mass".

If I have two positively charged objects at a distance r from each other, and then move them closer (say from r to r/2) the energy of the system doubles. The masses themselves did not gain more energy, the configuration did.

I'm also not sure what you mean by expressing these ideas mathematically, but this would likely require something like the calculus of variations, Lagrangians, and that sort of thing, which is quite advanced.

In short, kinetic energy is only affected by gravity if the force of gravity causes the object to accelerate (which obviously increases its kinetic energy). As far as the gravitational field itself, that contributes potential energy to the system, not kinetic.

Your question is rather confusing, but I hope I helped a little :)

anonymous · Answer

I'm guessing may be you are talking about the rest energy and rest mass of an object. Every  particle of rest mass m has associated with it rest energy, given by mc^2. Now all forms of internal energy (heat energy, potential energy of a compressed spring...) is reflected in the rest energy of the object. For example, a hot potato weighs more (has more rest mass) than a cold potato, and a compressed spring weighs more than a relaxed spring! Of course this difference is undetectable at normal energies since the square of the velocity of light is a very big number, and the change in rest mass is virtually undetectable. 
    When you add the kinetic energy you get the "total" energy. In the classical limit (low speeds and low interaction energies) the kinetic energy is approximately =(1/2) m v^2, which is the usual expression from mechanics.  
   I hope this sheds some light on your question...