# When moving from one position to another at a constant velocity, how does the conservation of energy hold?

I know that this might be a duplicate question, but I have not found any satisfactory answers that clear up my lack of understanding. Here is my question. Say a sloth hangs on a tree in the middle of nowhere. Therefore, he has some gravitational potential energy $mgh$. Next, say he falls to the ground. From the work-energy theorem, there was no net work done on him, since the his initial velocity was zero and his final velocity was zero. Therefore no energy was transferred, since there was no net work. However, now that the sloth is on the ground, there is no more gravitational potential energy, meaning that there is less energy in the system than there was initially. This seems to contradict conservation of energy. Obviously, I am making some form of conceptual error, but I am not sure how.

Conservation of energy, as you note, holds for "the system." For instance, if you push on a ball, that ball gains energy, but the energy of the ball is not conserved--only the energy of you and the ball. In this case, the system needs to include more than just "the sloth" because the sloth is not an isolated system--there are external forces at work. Here, the main one is the ground, which does work on the sloth. The sloth exerts a force on the ground, and some vibrations go off into the surrounding area, which are eventually damped out by friction and turn into heat.

What I'm saying here is that, if a sloth falls in the middle of the forest and nobody's around to hear it, it has to make a sound--by conservation of energy.

• Upvoting for that last sentence... Indeed if you imagine the sloth falls on a spring, you can see where the energy has gone (and where the work was done). – Floris Apr 21 '15 at 0:39
• I still don't understand. If kinetic energy and potential energy are the only "energies" in the system, it seems as though "the system," the sloth, the tree, and the earth has some potential energy that it does not have when the sloth is on the ground. But the total work in the system does not change because there was no change in velocity in the system, so it seems that the system loses energy. Are you simply trying to say that the gravitational potential energy is converted into sound and heat when the sloth impacts the ground? How would this apply if I raised the sloth? – Wesley Apr 21 '15 at 1:13
• I am indeed just saying that the sloth's energy becomes sound and heat. If you raised the sloth, you would do some work W to bring it up to the speed, and then you would have to do opposite work to bring it back to rest at its new, higher position. You wouldn't have to do as much slow-down work as speed-up work, though, because gravity would help you. The difference between the work you did to start the sloth moving and stop it moving is the gravitational potential energy the sloth has gained. – zeldredge Apr 21 '15 at 1:17
• Also consider that kinetic energy is heat. When the earth/atmosphere heats up because of the dissipated mechanical energy of the sloth, that is in fact a kinetic energy. It just doesn't show up if you think of things as point masses--the velocity of "the earth" doesn't change, but the velocity of little pieces of earth certainly does. – zeldredge Apr 21 '15 at 1:19
• @zeldredge Thanks. The falling down scenario makes perfect sense now. However, I am not as confident with the lifting one. By the end of it, the sloth has gravitational potential energy that the system did not have before since it was on the ground. Thus, where did this gravitational potential energy originate? Does it come from the chemical energy of me, the lifter? – Wesley Apr 21 '15 at 2:29

There is an external agent that removes mechanical energy from the sloth, namely the normal force exerted by the ground. You are right that no net work is done, but remember it is the work-kinetic-energy theorem: The net work equals the change in kinetic energy of the sloth.

When the sloth starts falling, it's potential energy starts getting converted to kinetic energy, and just before it hits the ground, all of its potential energy has been converted to kinetic energy. Now, the problem you're facing is because of the fact that you're considering just the sloth, which is wrong, because the sloth is interacting with the ground in this case, so with its kinetic energy, the sloth is hitting the ground, with some force. The energy the sloth has is utilized in its interaction with the ground, as in it gets converted to sound energy, heat energy and friction, and all the other forms of energy associated with the interaction. So the energy is conserved.