Very simple question regarding work and energy transfer

[Rayquesto]

potential energy plus the kinetic energy of a system is always the same, which can be accounted and assumed as zero. systems such as springs have a point in which the velocity is at its highest I think is called the eqaulibrium point. the points in which the spring reaches a maximum before stopping contains the highest potential energy and the highest force and highest acceleration. there's also the concept of work-kinetic energy theorem where work is transferred as kinetic energy is released, considering that velocity is constant. so, I hope you understand a little bit more. What are you confused about exactly?

 

[Rayquesto]

sorry not zero, but the kinetic energy plus the potential energy is constant.

 

[Dale]

One other problem with energy as the fundamental mechanism of injury is that energy is frame-variant and the extent of injury is not.

 

[Rap]

One other problem with energy as the fundamental mechanism of injury is that energy is frame-variant and the extent of injury is not.

I think this will not cause confusion. It will complicate the calculations, but not change the outcome. Just pick the simplest frame of reference (Earth stationary) and you can be certain that transformation to any other frame will modify the energy bookkeeping, but not the conclusions.

 

[Dale]

It will complicate the calculations, but not change the outcome.

That is just it. If energy were the sole cause of injury then because energy is frame-variant you would expect that injury would also be frame variant. But of course that is absurd, changing coordinate systems cannot possibly change the outcome, as you say. Therefore the inescapable conclusion is that injury is not merely due to energy.

Just pick the simplest frame of reference (Earth stationary)

Actually, I think that the best chance this idea has is to require that energy be defined in the body-stationary frame. In that frame the surface exerts a force on the body and as the body deforms it moves, so work is actually done on the body in this frame. Also, in the body-stationary frame the work done by a rigid surface is actually greater than the work done by a soft surface which corresponds with the greater injury. Of course, this frame is non-inertial.

 

[Rap]

That is just it. If energy were the sole cause of injury then because energy is frame-variant you would expect that injury would also be frame variant. But of course that is absurd, changing coordinate systems cannot possibly change the outcome, as you say. Therefore the inescapable conclusion is that injury is not merely due to energy.

Actually, I think that the best chance this idea has is to require that energy be defined in the body-stationary frame. In that frame the surface exerts a force on the body and as the body deforms it moves, so work is actually done on the body in this frame. Also, in the body-stationary frame the work done by a rigid surface is actually greater than the work done by a soft surface which corresponds with the greater injury. Of course, this frame is non-inertial.

I think that concluding that the injury is not due to energy is (practically) wrong. Its all a matter of bookeeping.

The bottom line is that, to do the problem very accurately, you have to use both conservation of energy and momentum. You have to assume that the Earth has finite mass.

If you use a frame where the Earth is initially at rest, then the only energies are the objects initial KE due to its velocity V, its final heat energy Q, its final KE (very small) and the Earth's final KE (very small). All the very small parts are not what we are interested in, so we can discard them. The momenta we have to deal with is the objects initial momentum, its final momentum (very small) and the Earth's final momentum. So that's easy, the initial momentum of the object gets transferred to the Earth and forget about the small stuff.

If you use an inertial frame where the object is initially at rest and the Earth is moving towards it at velocity V, then the energies are the initial KE of the object (zero), and the initial energy of the Earth (very large). The final energies are the KE of the object, the heat energy Q, and the KE of the Earth (very large). The final energies are the KE and Q of the object, and the KE of the Earth (very large). The initial momentum is just the momentum of the Earth (very large). The final momenta are the momentum of the object and the final momentum of the Earth (very large). Now we have a bunch of very large quantities that we are not interested in mixed in with the quantities that we are interested in.

This complicates things because now we have to watch the very large quantites closely, we cannot ignore them. We cannot assume the mass of the Earth is infinite, we cannot assume its initial and final velocities are the same. It complicates the bookkeeping and is not a good idea.

We definitely don't want to go to an accelerated frame of reference. Not only will we have all the problems listed above, but we will have to come up with a virtual force field to explain why inertial objects seem to accelerate.

 

[Dale]

Right, but in the Earth's frame energy transfer does not explain the injury since no energy is transferred (rigid surface). So despite the extra complications, you simply cannot use the "usual" frame if you want to equate energy transfer to injury.

Also, in biomechanics a body's rest frame, or even the rest frame of a single bone, is often used. So many people involved on the technical side of this field are familiar with non-inertial frames.

 

[Rap]

Right, but in the Earth's frame energy transfer does not explain the injury since no energy is transferred (rigid surface). So despite the extra complications, you simply cannot use the "usual" frame if you want to equate energy transfer to injury.

Also, in biomechanics a body's rest frame, or even the rest frame of a single bone, is often used. So many people involved on the technical side of this field are familiar with non-inertial frames.

Right, no energy is transferred from the Earth. It is the kinetic energy of the object which is transformed into heat as a result of the force applied by the Earth.

I think in this case we do not want to use huge numbers such as the mass of the Earth. If we stay in the body frame, we have to invent a force field which explains why the Earth decelerates from V to zero during the time of the collision. I don't think we want to do that either.

 

[Dale]

Right, no energy is transferred from the Earth.

This is the key point. There is an injury, there is no transfer of energy. Therefore, in the Earth's frame the transfer of energy is not the cause of the injury as asserted by the OP.

 

[Rap]

This is the key point. There is an injury, there is no transfer of energy. Therefore, in the Earth's frame the transfer of energy is not the cause of the injury as asserted by the OP.

Ah, I see what you are saying, and that is interesting. In the Earth inertial frame, there is no transfer of energy from the Earth, while in, let's say, the inertial frame of the object when it first touches the Earth, energy is transferred from the Earth to the object.

That means that the question of whether energy is transferred from the Earth is frame-dependent. I think that to do it properly, in any frame, the conservation of momentum must be included, so in that sense you are right, its not a matter of energy alone, its a matter of energy and momentum. Let me rephrase to say that the bookkeeping math for energy and momentum are simplest in the Earth inertial frame, where it happens that (practically) no energy is transferred from the Earth, and (practically) all of the object's initial momentum IS transferred to the Earth. Does that sound right to you?

 

[Dale]

That means that the question of whether energy is transferred from the Earth is frame-dependent.

Yes, exactly.

Let me rephrase to say that the bookkeeping math for energy and momentum are simplest in the Earth inertial frame, where it happens that (practically) no energy is transferred from the Earth, and (practically) all of the object's initial momentum IS transferred to the Earth.

Agreed.