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

[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?

 

[Doc Al]

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.

 

[Dale]

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.

 

[Cleonis]

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