Energy Change: PE to KE in Different Frames

[Dadface]

Imagine an event where, for example, two equal mass but oppositely charged objects approached each other from rest under the influence of the electrical force of attraction between them. In terms of energy we can say that the loss of electrical potential energy is equal to the gain of kinetic energy.

Now imagine this event being observed by two observers one at the centre of mass frame, the second in the non inertial frame of one of the accelerating particles. For simplicity let all velocities be low enough to be considered as non relativistic.

Now consider one point in the approach, where from the centre of mass frame each object has velocity v. From this frame the total KE of the system would be calculated as MV², where M equals mass of each object. From the non inertial frame the speed of approach of the second object would be seen as 2v and the KE calculated as 2Mv²

In other words the observed KE depends on the frame of the observer But shouldn't both observers agree about the change of PE? If so from energy conservation this implies the gain of KE should be the same for all observers. Obviously I'm overlooking something here? Any suggestions please?

 

[PeroK]

Imagine an event where, for example, two equal mass but oppositely charged objects approached each other from rest under the influence of the electrical force of attraction between them. In terms of energy we can say that the loss of electrical potential energy is equal to the gain of kinetic energy.

Now imagine this event being observed by two observers one at the centre of mass frame, the second in the non inertial frame of one of the accelerating particles. For simplicity let all velocities be low enough to be considered as non relativistic.

Now consider one point in the approach, where from the centre of mass frame each object has velocity v. From this frame the total KE of the system would be calculated as MV², where M equals mass of each object. From the non inertial frame the speed of approach of the second object would be seen as 2v and the KE calculated as 2Mv²

In other words the observed KE depends on the frame of the observer But shouldn't both observers agree about the change of PE? If so from energy conservation this implies the gain of KE should be the same for all observers. Obviously I'm overlooking something here? Any suggestions please?

You could take a simpler scenario where an object falls to the Earth. In the Earth's rest frame the PE is $mgh$.

What is the potential energy in a frame where the Earth and the object are initially moving at velocity $v$ in the direction of their separation?

 

[Dadface]

You could take a simpler scenario where an object falls to the Earth. In the Earth's rest frame the PE is $mgh$.

What is the potential energy in a frame where the Earth and the object are initially moving at velocity $v$ in the direction of their separation?

Thank you PeroK. I don't understand what you mean by "moving in the direction of their separation". Do you mean moving along a line which which is parallel to the line between the centre of mass of the object and earth? If so my initial feeling is that different observers would measure different separations. I need to think about it.

 

[PeroK]

Thank you PeroK. I don't understand what you mean by "moving in the direction of their separation". Do you mean moving along a line which which is parallel to the line between the centre of mass of the object and earth? If so my initial feeling is that different observers would measure different separations. I need to think about it.

I simply mean moving in the direction the object will fall. So that you have a 1 dimensional problem.

 

[Dadface]

Thank you. I thought the event in my scenario was one dimensional. Never mind.

This is how I see your scenario at the moment. In the frame of the Earth an observer will see the falling and accelerating object increasing its velocity. In the frame of the object an imaginary observer can describe that the Earth is increasing its velocity. In the centre of mass frame both objects can be described as increasing their velocity.

So as I see it the change of velocities and the resulting changes of KE calculated can be different for observers describing the same event but from different frames. What I can't see is why the change of PE can be different for different observers. I've got a feeling that it's due to non inertial frames and general relativity. Thank you.

 

[PeroK]

So as I see it the change of velocities and the resulting changes of KE calculated can be different for observers describing the same event but from different frames. What I can't see is why the change of PE can be different for different observers. I've got a feeling that it's due to non inertial frames and general relativity. Thank you.

In both cases the change in the Earth's velocity can be neglected as it is so small. For the second observer, both the Earth and the object about to be dropped are moving at the start.

The reason you can't see that PE is frame dependent is because you are unable to calculate the PE. It has nothing to do with General Relativity or non-inertial frames.

But,it does have to do with being able to calculate physical quantities in more than one reference frame. Have you ever studied a problem from more than one reference frame before?

 

[Dadface]

In both cases the change in the Earth's velocity can be neglected as it is so small. For the second observer, both the Earth and the object about to be dropped are moving at the start.

The reason you can't see that PE is frame dependent is because you are unable to calculate the PE. It has nothing to do with General Relativity or non-inertial frames.

But,it does have to do with being able to calculate physical quantities in more than one reference frame. Have you ever studied a problem from more than one reference frame before?

1. Ignoring the Earth's velocity would be a different scenario to what I described in my opening post. To make my scenario equivalent to yours let the "object to be dropped" have the same mass as the Earth (or a mass large enough so that the Earth's movement is not negligible)

2. In your second comment you seem to confirm that PE is frame dependent. I thought that was the case and I can't (yet) see why.

3. If I ever did study problems in more than one reference frame it was donkeys years ago and I can't remember it.

Thanks for your advice but it looks like I will need to go over some basics first so rather than waste your time I will try to look it up. I'm not even sure about the best sort of questions to google but I will try searching anyway. Thanks again.

 

[Dale]

@Dadface energy is definitely frame variant. In Newtonian mechanics KE is frame variant, but in relativity PE is also frame variant.

However, you definitely want to avoid non inertial frames. In general non inertial frames energy is not conserved.

 

[PeroK]

1. Ignoring the Earth's velocity would be a different scenario to what I described in my opening post. To make my scenario equivalent to yours let the "object to be dropped" have the same mass as the Earth (or a mass large enough so that the Earth's movement is not negligible)

I suggested the alternative scenario as it is simpler if only one object changes its velocity. It's slightly more complicated if both objects "move".

2. In your second comment you seem to confirm that PE is frame dependent. I thought that was the case and I can't (yet) see why.

If KE is frame dependent then PE must be too. You just have to do the maths, as they say.

Thanks for your advice but it looks like I will need to go over some basics first so rather than waste your time I will try to look it up. I'm not even sure about the best sort of questions to google but I will try searching anyway. Thanks again.

It is about the basics. In order to prove anything for yourself, you have to be able to set up the problem and solve it. Your question is a little obscure, as most problems on gravity work in the reference frame of a large mass that is assumed not to move.

But, a quick search found this:

https://www.physicsforums.com/threads/potential-energy-in-a-moving-reference-frame.64101/