Is Zero Acceleration Proof That an Object Must Be at Rest?

[Nile Anderson]

I think in trying to find a solution initially , others brought up debatable points

I'm not sure why this is still being debated, but 50 posts about a really badly worded question is definitely too many.

 

[sophiecentaur]

Here's where English or any language doesn't correlate to physical reality, Langauge is imprecise and there's no getting sound that.

Absolutely.
Maths seems to be the best available language that we have for describing physical situations because it is the most highly disciplined. Using 'words' always provides opportunities for misinterpreting questions and other peoples' answers. Mathsphobics beware.

 

[russ_watters]

That has NEVER worked for me during a test. Do the best you can was the typical answer...

Most of the time when kids ask, they are fishing. This time the question was actually badly written. I've had teachers correct a problem in the middle of a test.

 

[jerromyjon]

This time the question was actually badly written.

I have been in this situation several times. I cannot remember exactly what the questions were, but I have always been good at spotting contradictions. It more readily appears in multiple choice questions, where the "simple" answer that is expected (C) is competing with a more rigorously correct answer (D). Sometime in 7th grade I remember deciding to dumb myself down for tests. My math and chemistry teacher Mr. Glickman was scared to hand me and my best friend tests. We would compete to see who could find these "poorly worded questions" first. He actually allowed us to discuss things during tests because he knew we weren't cheating. We were finished with A's long before most other students were finished. The "book smart" ones would never beat us, because they were never taught the "tricks" we figured out ourselves.

Should, could or can instead of "must" resolves the conflict then "C" is the correct answer.

 

[pbuk]

Should, could or can instead of "must" resolves the conflict then "C" is the correct answer.

No, try my equivalent proposition:

• A wheel that is rotating should be on fire.
  Is there any reason that a rotating wheel should be on fire? No, the statement is false. Is the statement dependent on any particular set of circumstances? No, the statement is always false.
• A wheel that is rotating could be on fire.
  Could a rotating wheel be on fire? Yes, the statement is true. Is the statement dependent on any particular set of circumstances? No, the statement is always true.
• A wheel that is rotating can be on fire.
  Can a rotating wheel be on fire? Yes, the statement is true. Is the statement dependent on any particular set of circumstances? No, the statement is always true.

So what wording can produced the "desired" answer of C?

• A wheel that is rotating is on fire.
  Is it possible that a wheel can be both rotating and on fire? Yes, the statement can be true. Is the statement dependent on any particular set of circumstances? Yes, the statement is only true when the rotating wheel that is the subject of the statement is on fire: the statement is sometimes true.

Note that the wording in the question "If a wheel is rotating it must be on fire" will never yield C as the correct answer while it starts with "If..."

 

[pbuk]

Or a more rigorous analysis: let A be the set of objects with zero acceleration and B be the set of objects at rest.

Statement: If the object has zero acceleration the object must be at rest
Translation: ∀x: x ∈ A → x ∈ B (1)
Counter-example: an object with constant non-zero velocity: x ∈ A is true and so the proposition asserts x ∈ B but x ∉ B so the proposition is false.
Conclusion: (1) is false.

Statement: An object that has zero acceleration is at rest.
Translation: This is ambiguous, it could either mean x ∈ A → x ∈ B (1) as before, or alternatively it could mean x ∈ A ∧ x ∈ B (2).
Example of (2) is false: x is an object with constant non-zero velocity.
Example of (2) is true: x is an object at (continuous) rest.
Conclusion: (2) is sometimes true.

 

[my2cts]

The statement is indeed never true.
The answer D is correct and C is false.
Even for an object with zero acceleration and at rest
the statement that the rest condition is implied by the zero acceleration condition is false.
"False implies "never true".
The teacher is wrong. What he perhaps _intended_ to ask was not what he actually asked.
You can quote me on this.

 

[Fredrik]

Just as I thought I was out...they pull me back in.

Logically, there's no difference between the sentences

If an object has zero acceleration, it must be at rest.
​

and

If an object has zero acceleration, it is at rest.
​

They both mean the same thing, "something implies something". Unfortunately it's not clear what that something is. Is it

If v'=0 then v=0.
​

or

If v'(t)=0 then v(t)=0.
​

where t has some specific value?

Let's just say that it's the former, to keep things as simple as possible. The sentence "if v'=0 then v=0" doesn't have a truth value. I'll use a simpler example to explain. Consider the sentence x²=1. It doesn't have a truth value, but it can be given one by an assignment of a value to the variable x. If we specify that x represents the number -1, then that truth value is TRUE. If we specify that x represents the number 39, then that truth value is FALSE.

When we write something like x²=1 without having previously assigned a value to x, the intention is usually that the sentence should be interpreted as a part of a slightly longer sentence. That sentence is $\forall x~~ x^2=1$. This sentence has a truth value only if it's clear from the context what the scope of the "for all" is.

Now let's return to "if v'=0 then v=0". The two obvious ways to assign a truth value to this sentence is a) to completely specify what v is, and b) to add a "for all" to the start of the sentence, and make it completely clear what the scope of that "for all" is. Regardless of which of these options we choose, we run into additional complications. If we choose a), then we must specify a coordinate system, and the truth value of the implication will depend on that choice. If we choose b), then we must specify the scope of the "for all", and the truth value of the implication will depend on that choice.

All these attempts to prove that D is the only right answer are deeply flawed because they ignore all of these issues. There is no way to argue logically that D is the right answer, or that C is the right answer.

 

[sophiecentaur]

Most of the time when kids ask, they are fishing. This time the question was actually badly written. I've had teachers correct a problem in the middle of a test.

Been there myself - and delivered an apology to an A Level class for confusing them. Still remained friends with them, though. It's a different matter if a class is after your blood from the start, though.

 

[Doc Al]

I'm not sure why this is still being debated, but 50 posts about a really badly worded question is definitely too many.

Yes, I think this one has been beaten to death.