B Where does the 1/2 in 1/2 at^2 come from?

[topsquark]

Folks, I'll start this by saying I'm on a new medication and that I'm a little b!tchy. But please consider the following:
1. The OP requested a non-Calculus proof for the formula. This implies that they either have not taken Calculus or that they are in a non-Calculus course.

2. Mentioning that Calculus makes Physics easier is fine. Saying that it is a requirement in order to solve this problem is simply not true.

3. Telling the OP that you need Calculus to understand where the equation comes from is wrong, irrelevant to the OPs needs, and somewhat elitist. I totally agree that Calculus is an essential tool for understanding Physics. But it is not what everyone needs.

Please consider that this whole conversation about whether Calculus is necessary to understand Physics, or that Calculus makes it easier, is completely outside of the question posed by the OP. There is already another thread in progress about "how can you teach Physics without Calculus." I recommend that you put your arguments there, where it is more to the point, rather than here, where it is simply not relevant.

-Dan

 

[nasu]

If the OP needs an explanation without using some tool, the need is supposed to make it happen even if there isn't one?

He can get a justification, an intuition for why may be so, for why is reasonable to accept it for now as it is. Is done all the time in high school physics and algebra based university courses. What is the harm in telling him that this is not a proof without calculus, just a proof that uses an easier to understand method or result of calculus.

Like, area under the graph of force versus distance is equal to the work of the force or area under the graph of velocity versus time is the distance travelled. By using this he can see why for uniform acceleration the average velocity is equal to the arithmetic mean of velocity and why isn't in general so. Or why the work of the elastic force is the way it is.

But this not "without calculus", just without the word "calculus". Kind of politically correct physics in algebra based courses where the students get offended if they hear "calculus".

 

[topsquark]

If the OP needs an explanation without using some tool, the need is supposed to make it happen even if there isn't one?
He can get a justification, an intuition for why may be so, for why is reasonable to accept it for now as it is. Is done all the time in high school physics and algebra based university courses. What is the harm in telling him that this is not a proof without calculus, just a proof that uses an easier to understand method or result of calculus.
Like, area under the graph of force versus distance is equal to the work of the force or area under the graph of velocity versus time is the distance travelled. By using this he can see why for uniform acceleration the average velocity is equal to the arithmetic mean of velocity and why isn't in general so. Or why the work of the elastic force is the way it is.
But this not "without calculus", just without the word "calculus". Kind of politically correct physics in algebra based courses where the students get offended if they hear "calculus".

But the derivation does not require Calculus! The OP didn't want a Calculus based explanation, and it did not need one. My reply essentially comes from my High School Physics class, which was not an AP course (my High School didn't offer AP.) It is correct in all the details, and doesn't require any Calculus to understand it.

Yes, the proof is 4 lines shorter if you use Calculus. So what? The question is not about teaching the OP about areas under the curve nor is it about nor why the arithmetic mean of the velocity is not the same as the average velocity in the general case. When I presented my proof I did mention that the arithmetic mean equation only worked for constant acceleration. But by insisting on mentioning Calculus you are moving directly away from what the OP was asking: How to show where the 1/2 comes from without using Calculus. You aren't answering the question: you are expanding on it. That is simply not necessary in this case and it can only serve to put off the OP.

Yes, some people are threatened by Calculus. I don't really know why. Personally, I found my Calculus classes to be relatively easy. (So easy that I effectively taught myself most of Calculus I - III while I was in my Senior year of High School, mistakenly thinking that the whole text was just Calculus I.) But the fact remains that some people are. So why pressure the point? If the student is not bound to be a Physics, or related, major why worry about if they are not told a Calculus based proof when a non-Calculus based proof was readily available?

Just to be clear: whereas I am saying that it is possible to show where the 1/2 comes from without Calculus, I am generally not against your comments as to how Calculus is needed in learning Physics properly. It's just that it doesn't sound like the OP needs to learn those lessons, thus I do not believe that they belong in this thread: the Calculus comments are a tangent and do nothing to help the OP.

-Dan

 

[hutchphd]

the Calculus comments are a tangent and do nothing to help the OP.

If the OP thinks that his statement of the problem does not already use calculus he needs to be educated. I presume this need is why he has self-identified as a student. The very notion of an instantaneous velocity is the crux: an educator needs to be certain that knowledge is part of his Physics education despite the demand of "no calculus". The only question is how advanced.
I wish to learn English Literature but no Shakespeare.......
American History but no slavery please. I am weary.

 

[Vanadium 50]

I think we're getting tangled up in knots here.

Let's review - the OP asked where the 1/2 came from, and was told that the distance traveled is the average speed times the time. The average is half the sum of the initial and final velocities.

This is a fine, non-calculus based answer. Unfortunately, the OP rejected it.

Now we're trying to defend it by coming up with some sort of hand-wavy crypto-calculus. Given that we will never get to the bottom of the "but why?" ladder, nor can we guess what answer the OP finds acceptable, maybe we should have stopped there.

 

[Astro-Eddie]

Yes, v is the final velocity.

As to the average, for example, how do you find the average of the set
(v,t) = {(2.0,0), (2.5,1), (3.0,2), (3.5,3), (4.0,4)}?

That would be the middle point of the set, so
$\left ( \dfrac{2.0 + 4.0}{2}, \dfrac{0 + 4}{2} \right ) = (3.0, 2)$ (in whatever units.)

But this means that the average v is just the sum of the two endpoints divided by 2.

-Dan

Thank you so much!

 

[topsquark]

If the OP thinks that his statement of the problem does not already use calculus he needs to be educated. I presume this need is why he has self-identified as a student. The very notion of an instantaneous velocity is the crux: an educator needs to be certain that knowledge is part of his Physics education despite the demand of "no calculus". The only question is how advanced.
I wish to learn English Literature but no Shakespeare.......
American History but no slavery please. I am weary.

Where, exactly, did the OP ask about instantaneous velocity? And at what point are we designated as the educators in his/her course that we should teach Calculus in what is obviously a non-Calculus Physics course?

The OP asked a specific question. That question was answered in the manner that the OP asked for. Why are we supposed to add "...but the OP needs to know that Calculus is necessary to learn Physics" when the OP has clearly asked for a non-Calculus explanation, and say (incorrectly) that Calculus is necessary to understand where the 1/2 comes from?

To continue your example, if someone is asking a question about "The Metamorphosis" do we really need to say that Caliban wanted to rape Miranda so that he could create a race of Calibans on his island? Or that Prospero stopped him? Or, if someone is asking a question about the Maginot line, do we need to say that the Emancipation Proclamation was not what ended slavery in the US and that it was actually the 13th Amendment that legally made slavery a crime?

If someone asked a question about the force exerted by one charged particle on another, do we need to tell them that the Coulomb force is conservative and that they need to know how to prove that? Why not just answer the question?

Honestly, I'm not trying to tell anyone what they can/should say in a thread: I'm not Staff and I don't set the site policies. But the whole conversation about the need for Calculus in order to fully understand Physics is completely extraneous to the question that was asked. Are we here to answer questions that were posed or, seriously, are we here to tell the members that they don't know enough to answer questions that they didn't ask about? Sometimes, I agree, that comment does need to be made, but it did not need to be made here.

-Dan

 

[vanhees71]

I think the teaching of physics should also provide an idea, where the formulae come from you use to solve problems. In fact it's the real subject of physics. It's not physics to put numbers in a given formula, which you don't understand.

The very notion of the kinematical quantities needs calculus, whether you are allowed to call it calculus or not. The "calculus-free approach" is making things much less transparent, and I've no clue where the idea comes from that one should avoid it at all costs. As stressed several times, the notion that position-vector components can be calculated as the area under the space-time diagram graph (up to an additive constant, which is given by the initial conditions) already needs a heuristic idea of calculus. Then of course for $v_j=a_j t$ with $a_j=\text{const}$ this area of a triangle is given by the elementary geometric definition. The point is to understand why that's the case, and for this you need a heuristic understanding of calculus/integration.

 

[PeroK]

I think the teaching of physics should also provide an idea, where the formulae come from you use to solve problems. In fact it's the real subject of physics. It's not physics to put numbers in a given formula, which you don't understand.

The very notion of the kinematical quantities needs calculus, whether you are allowed to call it calculus or not. The "calculus-free approach" is making things much less transparent, and I've no clue where the idea comes from that one should avoid it at all costs. As stressed several times, the notion that position-vector components can be calculated as the area under the space-time diagram graph (up to an additive constant, which is given by the initial conditions) already needs a heuristic idea of calculus. Then of course for $v_j=a_j t$ with $a_j=\text{const}$ this area of a triangle is given by the elementary geometric definition. The point is to understand why that's the case, and for this you need a heuristic understanding of calculus/integration.

Where does calculus come from? If you insist on teaching calculus before you can calculate the area of a triangle or study conic sections - we are dealing with a parabola here - then someone else might insist on a course in real analysis before you can study calculus. What's the point of using integration if you can't prove where the formulas came from?

 

[vanhees71]

I do NOT insist of teaching calculus first. I insist on giving a heuristic explanation! Of course you start with the definition of the area of rectangles as (width times hight) of plane geometry. Then you need the idea of calculus to calculate the area under the graph of curve by first defining it approximately by the sum of areas of rectangles, which you make ever smaller to get a more and more accurate value for this area.

To get from $v_j$ to $x_j$ (up to an additive constant) again you need that $v_j=\dot{x}_j$, defined by the limit of finite differences $\Delta x_j/\Delta t$ for $\Delta t \rightarrow 0$. Then you can get the other way by approximating $\Delta x_j$ by $v_j \Delta t_j$ and sum over these rectangles to see that $x_j$ is (up to an additive constant) given by the area under the velocity-time diagram. Then, of course, if you know this area from elementary geometric considerations like the triangle for uniform acceleration, then you can write down this area without using formal integrals.

This kind of heuristic arguments are even a better approach to the application of calculus in physics than to just teach it as analysis a la Bourbaki. Of course, the serious theoretical physicist also needs this formal approach if it comes to the finer details, but at the level of high-school physics that would of course be overdoing it. Even in the math classes you get only to the most simple beginnings of the rigorous approach, as using the "$\epsilon$-$\delta$ definition" of limits, continuity, differentiability, and all that.

 

[haushofer]

Does she still love you?

Good thing she didn't ask you to optimize something...

Yes. My mother doesn't differentiate between me and my brothers.

 

[topsquark]

If the student was in a Calculus based class, then I would agree that teaching limits, area under the curve, etc. makes sense. The OP clearly is not in such a class. (I mean, not even the Calculus proponents felt the need to mention that $x = x_0 + v_0 t + (1/2) a_0 t^2 + \dots$ is the first few terms of a Taylor (technically Maclaurin) expansion, which is a more direct approach to the 1/2 coefficient than talking about areas under the curve.) Would you insist on speaking to a resident of Venice in Latin, as opposed to his language of Italian? All I am saying is that we need to talk to the student in the language they understand. We are not here to teach the OP their whole class, just to answer the question that was asked. Some extra information is fine, but insisting that the question cannot be answered without a knowledge of Calculus is not only a bit elitist, but actually wrong. (Those that disagree should probably go back and take another look at Pythagoras, the man who invented triangles. )

Please let me just say one last time that if someone is asking a question on the site, please focus on the following when answering:
1. If possible, speak the language the OP will understand.

2. If possible, answer the question directly. If it does require a bigger outlook than the student has presented then, yes, go ahead and say that. But if it does not, consider that the bigger outlook might do nothing but confuse the student.

3. If there is a simple way to answer the question, that's likely to be the better way of doing it, unless it's clear that the student will benefit from the more general approach.

Most that come here are likely to be Physics (or related) majors and will want a more general treatment. On the other hand, not everyone that does come here is a Physics major or even wants to be one.

Okay, I'm going to leave the conversation because some of you simply will not listen to what I think is reason. I'm sure that those same some of you feel the same way about me. In any event, however you look at it, a protracted argument on this does not serve anyone. The OP seems to be satisfied by the solution presented and I don't see that anything useful will happen from here on, so I'm done.

-Dan

 

[hutchphd]

you simply will not listen to what I think is reason

This is why we should listen to each other's ideas. It forms the fundamental purpose for an honest discussion.
I do find your accusations reflexive (as you infer before the fact). But making up one's mind before the discussion starts will likely shorten it.

 

[vanhees71]

If the student was in a Calculus based class, then I would agree that teaching limits, area under the curve, etc. makes sense.

You can't teach mechanics without these ideas. To avoid to call this calculus and the adequate mathematical language for formulating physics, is nonsense, because it makes the subject more difficult rather than simpler.

The OP clearly is not in such a class. (I mean, not even the Calculus proponents felt the need to mention that $x = x_0 + v_0 t + (1/2) a_0 t^2 + \dots$ is the first few terms of a Taylor (technically Maclaurin) expansion, which is a more direct approach to the 1/2 coefficient than talking about areas under the curve.) Would you insist on speaking to a resident of Venice in Latin, as opposed to his language of Italian? All I am saying is that we need to talk to the student in the language they understand. We are not here to teach the OP their whole class, just to answer the question that was asked. Some extra information is fine, but insisting that the question cannot be answered without a knowledge of Calculus is not only a bit elitist, but actually wrong. (Those that disagree should probably go back and take another look at Pythagoras, the man who invented triangles. )

Please let me just say one last time that if someone is asking a question on the site, please focus on the following when answering:
1. If possible, speak the language the OP will understand.

If you want to learn physics you have to learn the language that is used to talk about it. If you don't want to learn, we can't help.

2. If possible, answer the question directly. If it does require a bigger outlook than the student has presented then, yes, go ahead and say that. But if it does not, consider that the bigger outlook might do nothing but confuse the student.

3. If there is a simple way to answer the question, that's likely to be the better way of doing it, unless it's clear that the student will benefit from the more general approach.

You should make things as simple as possible but not simpler. One should be aware that a topic is remembered best in the way it is taught first, and it is difficult to unlearn wrong explanations. Fortunately also didactics makes some progress with the years. The invention of "calculus-free physics" by didacticians is a misguide!

 

[weirdoguy]

To avoid to call this calculus and the adequate mathematical language for formulating physics, is nonsense,

Stop calling what I, and others, do for a living a nonsense. I've also been thought that way in high-school, and there has been hundreds of exercises that were very difficult, and really using calculus wouldn't help. See for example physics olimpiads. You don't teach in high-school, so you simply do not know what you are talking about. You don't know what struggles students have, and 99% of them would not be resolved by using calculus.
Even I still sometimes have truggles with some of the problems, and I know calculus perfectly (I teach it also).I have an exercise for you: we have a rabbit that sits in the distance $d$ frome the road. There is a car on the road moving with the speed $u$. Rabbit wants to cross the road and the car is in the distance $l$ from the rabbit when rabbit starts to move. What is the minimal speed of the rabbit so that it won't be hit by a car? At what angle should it move?

Solve it without calculus. I would say that solving it this way requires a "little bit" more of understanding physics than using calculus.

 

[vanhees71]

I do not say that you cannot solve some kinds of problems without calculus. I say that physics gets unnecessarily complicated by avoiding the adequate language it is formulated in, which is calculus.

The problem is very vaguely formulated. Should the rabbit move in with constant velocity along a straight line? Then it's solving a system of linear equations with some parameters, which for given angle and speed of the rabbit admittedly is not calculus.

I don't say that there aren't very challenging problems which do not need calculus for their solution. I also know that teachers are obliged to follow study plans that are not optimal. In Germany we also have all kinds of "experiments" in other subjects and in elementary school. The result is that 1/5-1/4 of the fourth-graders (end of elementary school in Germany) are not able to read and comprehend simple texts...

 

[hutchphd]

Solve it without calculus. I would say that solving it this way requires a "little bit" more of understanding physics than using calculus.

With respect, if you are attempting to make a point of logic here it is lost on me. Perhaps I need some help.
The point of my objection to the OP is not doctrinal. I have taught freshmen both calculus-based and not calculus-based courses. To not be allowed to utter the word "calculus" is silly. Much of the subject, regardless of semantics, deals with instantaneous rates of change. The tapdance required not to call it calculus is exhausting. Avoiding the tapdance does not require introduction of partial diff. eq. into the syllabis. Also for understanding the historical import of Newton in the scientific revolution a descriptive knowledge of where calculus fits is important. Call it fluxions if you must.

 

[Tom.G]

This thread has reminded me of a circular firing squid.

 

[hutchphd]

In the limiting case only.....