# "Motion is impossible!" claims modern Zeno

1. Jun 16, 2015

### Nathanael

Let me set up a situation:

A piano rests on a frictionless surface. I am standing next to the piano (on a frictional surface) and I claim that the following two statements prove it is impossible for me to move the piano:

(1) ... The kinetic energy of the piano is equal to the work I've done on it.
(2) ... I can't do work on the piano unless it is moving. (But, because of (1), I can't get it moving unless I do work on it. But I can't do work on it unless it is moving... ad infinitum)

"Therefore the piano is immovable," I claim.

(1) is equivalent to the work-kinetic-energy theorem $W=\Delta E$
(2) is a special case (where $\frac{d\vec s}{dt}=0$) of the definition of work $dW=\vec F\cdot d\vec s=\vec F\cdot \frac{d\vec s}{dt}dt$

Please explain where and why my logic is flawed (assuming I don't know newton's laws).

2. Jun 16, 2015

### A.T.

The assumption of a prerequisite relationship between two quantities which change simultaneously.

3. Jun 16, 2015

### Staff: Mentor

Push it. You'll both move.

4. Jun 16, 2015

### Nathanael

Sounds reasonable, but I have to say I'm not satisfied (probably because I don't entirely understand you.) Can you say more about this?

Well, I probably won't move (I said my ground was frictional) but yes, it will move... but the only reason you know this is from F=ma.
But if you consider only work and energy you might come to the paradox I proposed.
I was curious if there is any way around this paradox without invoking F=ma.

5. Jun 16, 2015

### CWatters

You can apply a force on the piano without doing work (the piano is stationary).

If there is no friction to counteract the force you apply the piano MUST accelerate.

Once it's moving you can doe work on it.

6. Jun 16, 2015

### A.T.

When you have a quantitative relationship between two variables like y = x2, then changing one of them implies changing the other. If you misinterpret this correlation as a two way causation, you get: A change of y requires a change of x which requires a change of y... so no change of either is possible.

7. Jun 16, 2015

### Nathanael

Let me say, I do know that it will move. I do not believe my claim in the OP. (Perhaps I should have emphasized this.)

But my question is this:
If I didn't do work on it then how did it get to be moving?

I understand from the F=ma perspective: I push on it therefore it accelerates.

But from an energetic perspective: it must have energy (motion) before I can give it any energy.

8. Jun 16, 2015

### Nathanael

I feel like you're getting at the heart of it. Yet... it still doesn't resonate with me in this particular situation.
Somehow I can't get past this statement: "it must have energy (of motion) before I can give it any energy."

I will sleep on your explanation. Any different ways of explaining it are welcome. Goodnight.

9. Jun 16, 2015

### Staff: Mentor

Does it have to be moving, or does it just have to move?

10. Jun 16, 2015

### Nathanael

Sorry, I can't sleep. What you say sounds nice, but I can't understand how it is relevant:
We don't have something like y=x2, we have $\frac{dE}{dt}=k\sqrt{E}$ with E(0)=0. If E=0 then dE/dt=0 therefore the energy is in a steady state w.r.t. to time (i.e. it can't change).
($k=F\sqrt{\frac{2}{m}}$ not that it matters.)

I see what you're trying to say, but tell me if this is true or false: The only way to move is to, at some point in time, be moving.

[edited because I said "relative" instead of "relevant" ... I was tired haha]

Last edited: Jun 16, 2015
11. Jun 16, 2015

### Staff: Mentor

At some point in time, but not every point in time.

12. Jun 16, 2015

### jbriggs444

For every real number x greater than zero there is a real number y that is between x and zero. For every time when the piano is moving there is a prior time when the piano was moving.

One key to your conundrum is that you seem to want to imagine a first real number greater than zero -- a first instant when the piano is moving. But there is no such thing.

13. Jun 16, 2015

### Staff: Mentor

So what? It sounds like you are falsely equating force and energy. f=ma does not necessarily require energy.

If you apply a 1N force to a 1kg object, starting at t=0, at t=0 the object is stationary, but accelerating at 1m/s and at that time the rate of expenditure of energy is 0. There is no conflict.

Last edited: Jun 16, 2015
14. Jun 16, 2015

### A.T.

This is exactly like y=x2 where at x = 0 you have dy/dx = 0.

Natural language is ambiguous and another reason for apparent paradoxes.

Last edited: Jun 16, 2015
15. Jun 16, 2015

### Jeff Rosenbury

Why can't the piano have zero kinetic energy without you doing work on it?

What you seem to have proven is that you and the piano can't have a fixed position and zero momentum. Perhaps that's true instead?

16. Jun 16, 2015

### thefurlong

Classically, this fails for the same reason Zeno's paradox fails. It confuses differentials with discrete steps. Think about it this way: at 0 seconds, there is still a force, f (which you are applying to the piano). Now, imagine that a differential of time, dt, has passed. Then, the piano will undergo a change in momentum, f*dt. This, after all, is the definition of force. Now that the momentum is nonzero, the instantaneous velocity is also non-zero (though infinitesimal). By definition, velocity is the rate of change of position, so now, the position will be f*dt*dt at time 2dt.

Now, if you are familiar with calculus, then to get real numbers instead of infinitesimals, you just need to integrate, and voila! The piano is now moving, and work has been done.

17. Jun 16, 2015

### jbriggs444

This imagining just pushes the problem back into infinitesimals. One is still imagining a situation where after infinitesimal time dt the piano has infinitesimal but non-zero energy. Infinitesimal work must have been done. All of which is true enough.

As you say, this confuses differentials with discrete steps. The unstated intuition is that there must be steps and that there must be a first step.

18. Jun 16, 2015

### Staff: Mentor

It got to be moving by accelerating, and acceleration requires non zero impulse not non zero work. The work thing is a "red herring".

19. Jun 16, 2015

### Tom_K

To be honest, I don’t really understand the “problem” but:

Maybe this can be resolved by considering that the displacement of the piano does not happen instantaneously when you apply a force to it. I am thinking along the lines of a mass suspended from a spring and the mass is at the top end of travel. It is instantaneously at rest, but it is under acceleration from the force of gravity. That is, there is a phase difference between the acceleration and velocity but that certainly does not prevent the gravitational force from doing work on the mass.

I think the piano on a frictionless surface will behave in a similar fashion, with inertia causing a delay between acceleration and velocity.

20. Jun 16, 2015

### Jeff Rosenbury

I agree with you Nathanael. An object with no kinetic energy cannot move -- ever. At least not under the physical model you have chosen. There are other solutions to this equation, but the trivial solution does not allow change.

In a more complete model, energy is better defined by Noether's Theorem. Plus the incompleteness theorem indicates that no particle can have zero energy. Time is not as simple as your model implies either I think. I thought these sorts of paradoxes led to the development of Quantum Mechanics?