# Kinetic Energy of Objects in System, in different frames of reference.

• Puma24
In summary, kinetic energy is frame-dependent, meaning that the velocity of an object depends on the reference frame used for measurement. This can lead to different calculations of kinetic energy in different frames, but the principle of conservation of energy remains true. Inelastic collisions, where some kinetic energy is converted into work, can help explain any discrepancies between frames. Ultimately, there is no inconsistency in the concept of kinetic energy when considering the principle of relativity of inertial motion.
Puma24
Hey!

So, as I understand, kinetic energy of a moving object is proportional to its velocity squared. So I'm wondering where these inconsistencies come from, and how they are resolved:

So, say two objects of mass M are travelling, with reference to a stationary observer, one in the left hand direction and one in the right hand direction, both at velocity V.

To the stationary observer, they would record the energy of each object as 0.5*M*V^2 correct? Giving a total kinetic energy of M*V^2.

Now, in the same system, but taken from the frame of reference of one of the moving object, it observes the other moving away from it at a velocity of 2*V, and hence it would see the kinetic energy of the system to be 0.5*M*(2V)^2, which I suppose simplifies to 2*M*V^2.

Where does this change in energy come from? Am I making some incorrect assumptions?

There's no inconsistency. Kinetic energy is frame-dependent--the speed of an object depends on the frame doing the measurements. Something stationary in one frame is moving in another.

Note that, even though different reference frames will disagree about the amount of kinetic energy they will each agree that it is conserved in an elastic collision. So energy is frame-dependent but conserved.

Puma24 said:
Hey!

So, as I understand, kinetic energy of a moving object is proportional to its velocity squared. So I'm wondering where these inconsistencies come from, and how they are resolved:

So, say two objects of mass M are travelling, with reference to a stationary observer, one in the left hand direction and one in the right hand direction, both at velocity V.

To the stationary observer, they would record the energy of each object as 0.5*M*V^2 correct? Giving a total kinetic energy of M*V^2.

As an earlier replier pointed out: the foremost principle here is the principle of relativity of inertial motion.

The velocity that you attribute to some object is frame dependent. What is physically interesting is the relative velocity between objects. In the example of perfectly elastic collision: take several inertial coordinates system, and calculate for each one the total kinetic energy before the collision and after the collision. For instance, you can take the coordinate system that is co-moving with the common center of mass of the moving objects, or the coordinate system that is co-moving with one of the objects.
In a perfectly elastic collision the total momentum before and after the collision is the same amount, and you will find that the total kinetic energy is the same amount before and after the collision just as well!

Or you can work out what happens in perfectly inelastic collision (such as a marble embedding itself in a lump of clay.) Using the principle of conservation of momentum you can work out the velocity of the marble-in-the-clay object after the inelastic collision. You will find that in all frames the change in amount of kinetic energy comes out the same. That change of kinetic energy is the work that is done upon the clay, deforming it.

Summerizing:
The concept of kinetic energy combines perfectly with the principle of relativity of inertial motion; no inconsistency arises.

More generally, it's relatively straightforward to prove formally that http://www.cleonis.nl/physics/phys256/quantity_of_motion.php" . (The link I just added is to an article on my website.)

Cleonis
http://www.cleonis.nl

Last edited by a moderator:

## 1. What is kinetic energy?

Kinetic energy is the energy an object possesses due to its motion. It is defined as one-half of the object's mass multiplied by the square of its velocity.

## 2. How is kinetic energy affected by different frames of reference?

Kinetic energy is a relative quantity, meaning it depends on the observer's frame of reference. In different frames of reference, the same object may have different velocities, and thus, different kinetic energies.

## 3. Can an object have kinetic energy in a stationary frame of reference?

Yes, an object can have kinetic energy even in a stationary frame of reference. This is because kinetic energy is a measure of an object's motion relative to its frame of reference, not necessarily its absolute motion.

## 4. How does the kinetic energy of a system differ from the kinetic energy of individual objects within the system?

The kinetic energy of a system is the sum of the kinetic energies of all the individual objects within that system. This means that the total kinetic energy of a system is greater than the kinetic energy of any single object in that system.

## 5. Is kinetic energy conserved in all frames of reference?

Yes, according to the law of conservation of energy, the total kinetic energy of a system remains constant in all frames of reference, as long as there are no external forces acting on the system.

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