relativistic speeds as potential energy

[lemur]

As I understand it, the speed of light can be infinitely approached but doing so requires enormous amounts of energy-input for relatively little speed-gains. So, to give a clear but surely inaccurate example, accelerating from 0.85C to 0.95C may require less energy than acceleration from 0.95C to 0.98C (apologies if this isn't accurate examples, but you get my point). So my question is when something is accelerating at relativistic speed with decreasing speed-gain, doesn't the energy still have to be conserved? So does that energy get released as the particle or object decelerates a relatively small amount from near-C speed? If so, is or could this be related to the changing energy-levels of electrons as they absorb and emit photons, since relatively large amounts of energy could be stored as relatively small increments of speed-change, which could be so small that they become quantized? I admit this is a grand conclusion to stretch from my initial question, and if necessary I am happy to repost in speculations - though I am still interested in the application of established knowledge to this question of energy-storage in relativistic speed-increases.

[swansont]

Energy is conserved. The kinetic energy is given by [math](\gamma - 1)mc^2[/math] where [math]\gamma[/math] is the same factor we use for time dilation and length contraction

[Sisyphus]

As I understand it, the speed of light can be infinitely approached but doing so requires enormous amounts of energy-input for relatively little speed-gains. So, to give a clear but surely inaccurate example, accelerating from 0.85C to 0.95C may require less energy than acceleration from 0.95C to 0.98C (apologies if this isn't accurate examples, but you get my point).

 

Yes. So for example an object at 0.99C has about 40 times the kinetic energy as an object at 0.5C.

 

So my question is when something is accelerating at relativistic speed with decreasing speed-gain, doesn't the energy still have to be conserved?

 

Yes.

 

So does that energy get released as the particle or object decelerates a relatively small amount from near-C speed?

 

What energy? It takes the same amount of energy to "decelerate" something from 0.98C to 0.95C as it does to accelerate it from 0.95C to 0.98C. There is no extra.

 

If so, is or could this be related to the changing energy-levels of electrons as they absorb and emit photons, since relatively large amounts of energy could be stored as relatively small increments of speed-change, which could be so small that they become quantized? I admit this is a grand conclusion to stretch from my initial question, and if necessary I am happy to repost in speculations - though I am still interested in the application of established knowledge to this question of energy-storage in relativistic speed-increases.

 

I don't really know what this means, so I'll just say to keep in mind that kinetic energy is relative. The object might be going at 0.98C in our rest frame, but there is also a reference frame in which its velocity is zero. There is no event that can occur at 0.98C that wouldn't happen for an object at rest, because it is also at rest.

[lemur]

What energy? It takes the same amount of energy to "decelerate" something from 0.98C to 0.95C as it does to accelerate it from 0.95C to 0.98C. There is no extra.

That's a good point, but then what happens to the energy put into accelerating the object once more energy is added to it to decelerate it? That energy has to be conserved so it has to go somewhere. When the object accelerates, the added energy is expressed as speed/momentum. So when it decelerates, you're basically saying that this is acceleration from the opposite direction as it accelerated (since it is at rest relative to itself). But since speed has to be measured relative to some other point, changes in momentum relative to that point constitute storing or releasing of energy, no? This is confusing because motion is typically identified as kinetic energy, not potential, but doesn't momentum store energy that is released upon collision with another object? So, if an object is collides with another at 0.98C, wouldn't the impact transfer more energy than at 0.95C according to how much energy was required to accelerate it from 0.95C to 0.98C?

 

I don't really know what this means, so I'll just say to keep in mind that kinetic energy is relative. The object might be going at 0.98C in our rest frame, but there is also a reference frame in which its velocity is zero. There is no event that can occur at 0.98C that wouldn't happen for an object at rest, because it is also at rest.

Isn't this like saying that there's no event that can occur in a car traveling 100mph that wouldn't occur if the car was parked? That would be true as long as the rest frame of the car doesn't interact with the rest-frames of any other cars, trees, etc. but if it did, the events inside the car would suddenly reflect the energy of the frame relative to the frame it's running into (or which is running into it, if you prefer to look at it that way)? But I think you're actually saying the same thing I'm saying with kinetic energy in one frame existing as potential energy within the rest-frame of the object.

 

Now my question is whether this has any relationship to energy levels of atomic electrons. I.e. do they accelerate to move away from the nucleus and by doing so acquire a type of positional potential energy? Does that also explain how ionization can eventually occur and why ions can be used to store electrical energy in a battery? Or am I confounding things with flawed basic assumptions?

 

 

Edited February 10, 2011 by lemur

[swansont]

That's a good point, but then what happens to the energy put into accelerating the object once more energy is added to it to decelerate it? That energy has to be conserved so it has to go somewhere. When the object accelerates, the added energy is expressed as speed/momentum. So when it decelerates, you're basically saying that this is acceleration from the opposite direction as it accelerated (since it is at rest relative to itself). But since speed has to be measured relative to some other point, changes in momentum relative to that point constitute storing or releasing of energy, no? This is confusing because motion is typically identified as kinetic energy, not potential, but doesn't momentum store energy that is released upon collision with another object? So, if an object is collides with another at 0.98C, wouldn't the impact transfer more energy than at 0.95C according to how much energy was required to accelerate it from 0.95C to 0.98C?

 

Energy and momentum are related but not interchangeable — they are two separate properties. Momentum does not "store energy." Sisyphus has already given an example; something moving at 0.98c has almost twice the kinetic energy as something moving at 0.95c. Since it has more energy it would tend to transfer more energy in a collision.

 

Now my question is whether this has any relationship to energy levels of atomic electrons. I.e. do they accelerate to move away from the nucleus and by doing so acquire a type of positional potential energy? Does that also explain how ionization can eventually occur and why ions can be used to store electrical energy in a battery? Or am I confounding things with flawed basic assumptions?

 

Potential energy is positional energy; charges that have a larger separation will have closer to zero potential energy. In the case of opposite charges this is a larger value, since PE is negative.