[Physics] Newtonian mechanics – where does the energy come from

Question

I am imagining an object on the floor. If I pick up the object moving it upwards with a constant velocity then a net force must be applied to that object in order to have it go from not being in a state of rest to being in motion. It has accelerated and F(net)=ma. It now has kinetic energy equal to 1/2mv^2. As the object moves upwards (at constant velocity) it gains gravitational potential energy (GPE). I am under the impression that the work done on that object is equal to the GPE that it gains as it moves upwards.

So if the work done has gone into storing the GPE and they are equal, where did the energy come from which made it accelerate and gain kinetic energy in the first place? It seems as though the energy put into lifting it must be equal to the acceleration caused and thus the kinetic energy gained AND the GPE too. I am also under the impression that the kinetic energy and GPE are equal. So it seems as though I have done work on the object and the work done is equal to the sum of K.E and GPE. How is that possible? Surely that way I have gotten out twice the energy that I put in which is clearly nonsense.

Also, if the K.E and GPE are equal, how is it that it can keep moving upwards with a constant velocity and hence a constant K.E and the GPE is increasing?

The more I write and the more I go over this in my head, the more confused I get and the more frustrated I get which just makes it more difficult. Not to mention the headaches!

Again, sorry if this is really stupid and i'm fairly certain that what I have asked here will make little sense but it's the best that I can do with the mess in my head. Even just writing this I have thought of many other apparent problems and contradictions which I can't even begin to formulate into a coherent question.

Any help that anybody can offer will be greatly appreciated.

Answer 1

The main equation for you is that of energy conservation:

$$K_1+U_1+W=K_2+U_2$$

The numbers represent before and after, or situation 1 and situation 2.

• Kinetic energy $K$ is "motion" energy. Objects having a speed (not acceration, that doesn't matter) have kinetic energy: $$K=\frac12 mv^2$$
• Potential energy $U$ is "stored" energy. This appears when objects "want to" move somewhere else - in this case the object wants to fall down again because of gravity, so by lifting it up there is "stored" energy which can come in use by letting it go: $$U=mgh$$
• Work $W$ is energy added. And work is done by forces $F$: $$W=F x$$ where $x$ is the distance moved. So, if a force - like the one applied by your hand when lifting the object - lifts the object the distance $x$, then it has done the work $W=Fx$ on the object; in other words, it has added this mount of energy to the system.

To begin with there is no motion and the object is at the bottom, so $K_1=0$ and $U_1=0$. You then do work $W$, which turns into kinetic energy while some of it is also stored as potential energy. The equation becomes:

$$K_1+U_1+W=K_2+U_2 \quad\Leftrightarrow\quad 0+0+W=K_2+U_2 \quad\Leftrightarrow\quad Fx=\frac12mv^2+mgh$$

So the work is equal to the sum of the final energies in situation two. All that energy comes from the energy input in the form of work.

The acceleration does not have an influence on all this. The only thing acceleration has an influence on is how big the force will be - from Newton's 2nd law, a larger acceleration requires larger forces. So if the object is accelerated largely, then that would have caused a larger force and thus more work $W$ done.

Even just writing this I have thought of many other apparent problems and contradictions which I can't even begin to formulate into a coherent question.

Don't give up!