Would this method still work in an expanding universe?

[tony873004]

On a test, I was marked wrong on the following 2 questions.

If you were the teacher, would you have marked me wrong?

1. State the definition of the 1st derivative.
My answer:
Function that gives the slope of the tangent line for any value of x that exists.
My score on this question: 0/2

The answer the teacher wanted:
<br /> f&#039;(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}<br /> <br />
Obviously, although he asked for the definition, he wanted the mathematical definition, and gave no points to anyone who gave a word-based definition.

2. Suppose and object is thrown down from the top of a building at 100 ft/s. The building 1100 feet high. Its height s(t) after t seconds is given by s(t) = 1100-100t - 16t².
a. Find the object's average velocity on the interval [1, 3]
What I did:
s'(t)=-100-32t
s'(1)=-100-32(1)=-132
s'(3)=-100-32(3)=-196

\frac{-132+(-136)}{2}=-164 ft/s
My score 1.5/2

What he wanted:
<br /> \frac{s(3)-s(1)}{3-1} = <br /> <br /> <br /> \frac{1100-100(3)-16(3)^2-(1100-100(1)-16(1)^2)}{3-1}) = -164 ft/s<br /> <br />
There were certain problems on this test where he told us that we may not use certain methods (i.e. no power rule allowed on certain problems), but he did not specify a method here.

 

[Galileo]

I would've marked them the same way.

The first question clearly asked for the definition and the limit definition is the only rigorous one. The geometric interpretation of a derivative is different from its definition. And a function is not the same as the graph of that function. (Some functions are quite ungraphable).

Your answer to the second question is correct, but you didn't state why you could use the method you did. The definition of the average velocity over some interval [a,b] is
\frac{s(b)-s(a)}{b-a}

Only when the acceleration is constant can you say that the average velocity over the interval is the average of the velocity at the start and the velocity at the end:
\frac{s&#039;(a)+s&#039;(b)}{2}. Since you didn't state that the acceleration was constant you didn't provide a complete solution. That's why you got 1.5 points out of 2, which is fair. I would've graded it the same way.

 

[tony873004]

But acceleration is constant as given by the question:

s(t) = 1100-100t - 16t2

He could have given us a longer formula that contained things such as G, M_Earth, R_Earth, but that would have been beyond the scope of this class. The formula, as he gives it implies a constant g.

Even so, if we were to use a non-constant g, then his method fails too.

I had a quick 10 seconds to ask him about this after class (many others were waiting to ask questions too), and he said I was marked wrong because he asked for an average, and by finding the first derivative, I was finding an instantaneous velocity. His argument made no sense to me. On Monday I get a chance to spend more time to argue my point. So I appreciate you (and anyone else who cares to participate in this thread) arguing the teacher's side for me.

btw... I don't want to give anyone the impression that I dislike my teacher. He's one of the 5 best teachers I've had in my life, and a few other students agree. He explains things clearly, or I'd be visiting this sub-forum a lot more often.

 

[0rthodontist]

I think the second one should get full credit though the first one shouldn't get any credit. It is the case that your method works under constant acceleration and the problem concerned constant acceleration. If he wanted to mark you wrong for not understanding average velocity, he should have done so on another problem where misunderstanding would have led to a wrong answer. As it is, he has no case that you didn't completely understand everything and neglect to mention the fact of constant acceleration simply because you took it as implicit.

 

[Tide]

FWIW, I concur with Galileo.

 

[AKG]

You should have gotten 0 on the first. On the second, you were lucky to get 1.5/2. You didn't prove acceleration was constant. Also, you didn't prove that when acceleration is constant, the average velocity over an interval is the arithmetic mean of the initial and final velocities.

 

[tony873004]

You should have gotten 0 on the first. On the second, you were lucky to get 1.5/2. You didn't prove acceleration was constant. Also, you didn't prove that when acceleration is constant, the average velocity over an interval is the arithmetic mean of the initial and final velocities.

I wasn't asked to prove that stuff. I was asked to find the average velocity. I found the average velocity. My answer precisely agreed with the the answers of the students who were given full credit. We were not prohibited from using any methods. Both methods need to assume a constant acceleration. If they didn't, then we would have needed to be given a MUCH more accurate value than 16 (half or 32 ft/s) in the original formula or the effects of a non-constant g would be completely drown-out by a value of g given by 2 significant digits.

FWIW, I concur with Galileo.

I'm curious why since Galileo's reasoning was due to a non-constant g, and the teacher's method fails with a non-constant g as well. (btw... this is not an astrophysics class).

And I have 1 more question to put before you all... I'll put it in the next post to give me time to work on my TEX.

 

[tony873004]

Calculate the average rate of change for the indicated value of x.
f(x)=x²-5; x=3

Since we were not prohibited from using the power rule, my answer was:

f'(x)=2x
f'(3)=2*3=6

My score: 1.25/2

What the teacher wanted is:

6+h.

My reasoning as to why he's wrong:

6+h would be the average rate of change for [x, x+h], but not for x.
x is a single point. It's like asking what's the batting average of a player who has 1 hit in 1 at bat. It's 1/1. Here too, with only a single point, the denominator of a fraction designed to compute an average must contain a number n=number of data points.

So the average rate of change for a single point should be the instantaneous value of that point.

Again, thanks to everyone for playing the part of the teacher in debating against me.

 

[neutrino]

Also, I hope you didn't write down 196 as 136 in your paper.

 

[tony873004]

Also, I hope you didn't write down 196 as 136 in your paper.

Well at least you're paying attention That was a typo in the middle of the problem that doesn't work its way into the answer. And, no, I didn't make a typo on my test :).

 

[Galileo]

I wasn't asked to prove that stuff. I was asked to find the average velocity. I found the average velocity. My answer precisely agreed with the the answers of the students who were given full credit. We were not prohibited from using any methods. Both methods need to assume a constant acceleration. If they didn't, then we would have needed to be given a MUCH more accurate value than 16 (half or 32 ft/s) in the original formula or the effects of a non-constant g would be completely drown-out by a value of g given by 2 significant digits.

I'm curious why since Galileo's reasoning was due to a non-constant g, and the teacher's method fails with a non-constant g as well. (btw... this is not an astrophysics class).

Well, as unfair as it may sound afterwards, the obtained answer is not the most important thing. The reasoning behind it is. The journey is more valuable than the destination (or something like that).

The average velocity is by definition the displacement divided by the time interval:
\bar v = \frac{\Delta s}{\Delta t}=\frac{s(b)-s(a)}{b-a}
This is (by definition) general. It always holds. Constant acceleration or not.
By constrast, if the acceleration is constant, there is another way. he velocity graph is a straight line, to find the average you can simply take the average of the end velocities. But this is not general. It doesn't holds for, say s(t)=t^3 from t in [0,1].

Therefore, your answer is incomplete without specification that the acceleration is constant, which you could show by calculating s''(t)=-32 (=constant). So in all fairness, you got a little bit less than the full 2 points. 1.5 is not bad, right? Consider you learned from it.

 

[Curious3141]

Well, as unfair as it may sound afterwards, the obtained answer is not the most important thing. The reasoning behind it is. The journey is more valuable than the destination (or something like that).

The average velocity is by definition the displacement divided by the time interval:
\bar v = \frac{\Delta s}{\Delta t}=\frac{s(b)-s(a)}{b-a}
This is (by definition) general. It always holds. Constant acceleration or not.
By constrast, if the acceleration is constant, there is another way. he velocity graph is a straight line, to find the average you can simply take the average of the end velocities. But this is not general. It doesn't holds for, say s(t)=t^3 from t in [0,1].

Therefore, your answer is incomplete without specification that the acceleration is constant, which you could show by calculating s''(t)=-32 (=constant). So in all fairness, you got a little bit less than the full 2 points. 1.5 is not bad, right? Consider you learned from it.

I'd like to add the definition for the accepted meaning of "average value of a function".

Say you have a function f(x) defined and integrable over an interval [a,b]. The average value of that function over that interval is defined as :

f_{avg} = \frac{1}{b-a} \int_{a}^b f(x)dx

That applies for *any* function. If you take the same equation and apply it to a problem of motion to calculate average velocity over the interval [t1, t2], you get :

v_{avg} = \frac{1}{t_2-t_1} \int_{t_2}^{t_1} v(t)dt

Keeping in mind that v(t) = ds/dt that becomes :

v_{avg} = \frac{1}{t_2-t_1} \int_{t_2}^{t_1} \frac{ds}{dt}dt

v_{avg} = \frac{1}{t_2-t_1} [s(t_2) - s(t_1)]

or, in other words, the total displacement divided by the total time. This method made no assumptions about the acceleration, and therefore can be applied in all circumstances.

You'll be seeing more of average value of a function in physics, notably in a.c. circuit theory.

 

[tony873004]

...or, in other words, the total displacement divided by the total time. This method made no assumptions about the acceleration, and therefore can be applied in all circumstances...

agreed, except that total displacement was computed assuming constant acceleration.

 

[Nimz]

To get full credit on that second problem, given your method, I would have expected something like, "since the acceleration is constant, average velocity is blah blah blah." If you applied your method to any old problem, it wouldn't give you the right answer in general. Since net displacement over total time is the definition of average velocity, that method will work every time.

If the formula given had variable acceleration, would you have attempted the same method? If I was your teacher, I might not know. Thus, without the disclaimer that the method only works in constant acceleration, I agree with your teacher and Galileo with the 1.5/2 score.

For the third problem, it isn't possible to find the average rate of change at exactly a point. There is no change to find an average rate of change over. Because the question specified an average rate of change, not an instantaneous rate of change, you are technically incorrect. Seeing as you had about half of the answer, the score your teacher gave seems fair.

 

[Curious3141]

agreed, except that total displacement was computed assuming constant acceleration.

It's getting a little confusing. Let's simplify things :

1) Average velocity = total displacement/total time *always* works regardless of whether acceleration is constant or variable. THe derivation of this formula (which I gave) makes no assumptions about the nature of the velocity function v(t) or its first derivative, the acceleration a(t).

2) Average velocity = (initial velocity plus final velocity)/2 *only* works given constant acceleration. Here an assumption is made that v(t) and a(t) fit certain conditions.

You're not wrong in using 2) to get your answer here. However, I think to be safe, you should've explicitly mentioned that you're aware that acceleration is constant here, which is why you're using this method. See, the problem is that a naive student may think in terms of "simple" averaging - arithmetic means - of two numbers, which basically entails adding them together and dividing by two. Your teacher may have gotten the impression you're thinking this way - that you can find average velocity by just adding and dividing by two. He/she might be afraid you'll carry this same method over to problems with varying accelerations (of course, that'd be wrong), so he/she thought it would be better to "nip the habit in the bud", sort of.

If you were very clear in your mind about what you were doing, and you wouldn't use the method you did if you were given the same problem with the equation s(t) = 2t^3 -6t^2 + 5t + 7 or s(t) = 5cos(t) + t^2 or something like that, then by all means, explain to your teacher that you know when you can use the method you used.

But to keep things simple, it's easiest to just use the total disp/total time method. Since that method is actually so much easier to use here, you should've used that.

In fact, if you don't mind my saying so, it seems weird that you used the method you used (taking the derivative and everything), it suggests that you actually have a misconception about the way average velocity should be worked out.

 

[matt grime]

1. When asked for a definition give one, don't give a description of an interpretation of something. Also, the description you gave doesn't actually make sense (for any value of x that exists?)

 

[AKG]

I wasn't asked to prove that stuff. I was asked to find the average velocity. I found the average velocity. My answer precisely agreed with the the answers of the students who were given full credit. We were not prohibited from using any methods. Both methods need to assume a constant acceleration. If they didn't, then we would have needed to be given a MUCH more accurate value than 16 (half or 32 ft/s) in the original formula or the effects of a non-constant g would be completely drown-out by a value of g given by 2 significant digits.

That's wrong. The way he computed average velocity does not depend, whatsoever, on what the acceleration is like. He uses the direct definition of average velocity: (Final position - Initial position)/(Time elapsed). All you have to do is be able to plug 3 and 1 into your formula for s(t). It doesn't matter if acceleration is 32, or e^t², you just need to plug in numbers and use a calculator. Your method needed to assume constant acceleration, his did not. Yes, you got the right answer, but you were lucky. His solution uses the definition of average velocity, whereas yours uses a formula that only works in special cases. Since you didn't prove that you were in this special case, why would you get full marks?

The most important thing isn't getting the right answer. A cheater could even do that. The most important thing is knowing how to get a right answer, and knowing why a given answer is right or wrong. You didn't demonstrate that you know why your answer is right, since you didn't demonstrate that you know why you can apply the formula you used.

EDIT: In fact, your comments above, saying that his method requires a more accurate acceleration shows that you don't fully understand what's going on. In fact, accuracy has nothing to do with anything here. 16 is not given as a measured value, where there might be room for error. It is given as an exact value.

 

[nrqed]

On a test, I was marked wrong on the following 2 questions.

If you were the teacher, would you have marked me wrong?

1. State the definition of the 1st derivative.
My answer:
Function that gives the slope of the tangent line for any value of x that exists.
My score on this question: 0/2

The answer the teacher wanted:
<br /> f&#039;(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}<br /> <br />
Obviously, although he asked for the definition, he wanted the mathematical definition, and gave no points to anyone who gave a word-based definition.

I agree with your teacher for one reason: your answer just leads to the obvious question: but how is the slope of a tangent line defined then? I know you will say the answer to that one is obvious, but it is not. When providing a mathematical definition of something to be calculated, it must be something that provides an unambiguous way to do the calculation. If you define in terms of something else, you must define this something else. Of course, at some point, one had to agree on what *is* already defined and what must be defined by you but it's clear in that context that 'slope of a tangent' is not clearly defined.

2. Suppose and object is thrown down from the top of a building at 100 ft/s. The building 1100 feet high. Its height s(t) after t seconds is given by s(t) = 1100-100t - 16t².
a. Find the object's average velocity on the interval [1, 3]
What I did:
s'(t)=-100-32t
s'(1)=-100-32(1)=-132
s'(3)=-100-32(3)=-196

\frac{-132+(-136)}{2}=-164 ft/s
My score 1.5/2

What he wanted:
<br /> \frac{s(3)-s(1)}{3-1} = <br /> <br /> <br /> \frac{1100-100(3)-16(3)^2-(1100-100(1)-16(1)^2)}{3-1}) = -164 ft/s<br /> <br />
There were certain problems on this test where he told us that we may not use certain methods (i.e. no power rule allowed on certain problems), but he did not specify a method here.

As Galileo said, your method works only for constant acceleration. You shoudl have said ''I may use this here because the acceleration is constant and I can tell this because...''.
The problem is that your teacher can not tell if you knew that this was essential to your approach or if you thought that you could always do that. He is not psychic ;-) The problem is that others could do it the same way and never understand that their calculation worked only because a was constant. And if the don't they obviously don't deserve full credit.

Hope this makes sense.

Patrick

 

[tony873004]

Thank you for all the replies so far. You're helping me anticipate what the teacher might argue when I try to convince him for a few points back.

The general consensus so far seems to be that I'm wrong about the gravity problem because my method needs constant acceleration to be valid. I agree with that, but so does his method. Let's try a different approach. I'm going to re-word the problem with different numbers, so a non-constant gravitational field does make a significant difference. I'm also going to switch to metric units:

Suppose an object is thrown down from the top of a building at 30.48009 m/s. The building is 4000 kilometers high. It's height s(t) after t seconds is given by s(t)=13123320-30.48009t-4.9t². Find the object's average velocity on the interval [1, 1200]

Accounting for a non-constant gravitational field, this object falls from r(center of Earth)=10378000 meters to r=7376000 meters (altitude, 4000km to 998km) in the 1200 second interval.

It's average velocity is (10378000-7376000)/(1200-1)=2504 meters / second.

Let's see how my method and how his method compare to 2504 meters per second.

My method:
s'(t)=(-100-9.8(1) + (-100-9.8(1200))) / 2 = -5915 meters per second

His method:
(13123320-30.48009*(1200)-4.9*(1200)^2 - 13123320-30.48009*(1)-4.9*(1)^2)/(1200-1)=-5915 meters per second.

We still get the same answer, and we're both wrong. His method NEEDS to assume constant gravity too.

I want to address some of the other points made in this thread, but I have to go to work now. Thanks again for all the replies.

 

[nrqed]

Thank you for all the replies so far. You're helping me anticipate what the teacher might argue when I try to convince him for a few points back.

The general consensus so far seems to be that I'm wrong about the gravity problem because my method needs constant acceleration to be valid. I agree with that, but so does his method.

No. This is where you are wrong (please reread AKG's post).

The definition v_{x average} = { \Delta x \over \Delta t} is a general definition, valid for any type of motion at all. The trick v_{x average} = {v_{x f} + v_{x i} \over 2 } is only correct for constant acceleration.

Let's try a different approach. I'm going to re-word the problem with different numbers, so a non-constant gravitational field does make a significant difference. I'm also going to switch to metric units:

Suppose an object is thrown down from the top of a building at 30.48009 m/s. The building is 4000 kilometers high. It's height s(t) after t seconds is given by s(t)=13123320-30.48009t-4.9t². Find the object's average velocity on the interval [1, 1200]

But you are still using an example with a constant acceleration! . It does not suffice to say "the acceleration is not constant", you must use am equation that reflects that! You are still using an equation for s(t) that corresponds to a constant acceleration of -9.8 m/s^2! So you are saying that you are looking at motion where the acceleration is not constant and then you start using an equation describing constant acceleration! So this is inconsistent!

Accounting for a non-constant gravitational field, this object falls from r(center of Earth)=10378000 meters to r=7376000 meters (altitude, 4000km to 998km) in the 1200 second interval.

It's average velocity is (10378000-7376000)/(1200-1)=2504 meters / second.

Let's see how my method and how his method compare to 2504 meters per second.

My method:
s'(t)=(-100-9.8(1) + (-100-9.8(1200))) / 2 = -5915 meters per second

His method:
(13123320-30.48009*(1200)-4.9*(1200)^2 - 13123320-30.48009*(1)-4.9*(1)^2)/(1200-1)=-5915 meters per second.

We still get the same answer, and we're both wrong. His method NEEDS to assume constant gravity too.

Of course, because you used the wrong equation for s(t)! (you used one that describes a constant acceleration!)

Look, consider instead s(t) = 1 + 2 t^3. Now calculate the average velocity between 2 and 4 seconds using both methods. They won't agree and the correct one will be the one given by the prof's approach.
So you were lucky to get the correct answer in the test, because if it had been a function not corresponding to constant acceleration, you would have got zero point!

Hope this clarifies things!

Pat

 

[tony873004]

Thanks, Pat. I don't quite understand a few of your comments. I did read AKG's post.

...The way he computed average velocity does not depend, whatsoever, on what the acceleration is like.

But it does. In the teacher's method he computes average velocity by computing 2 distances and using those two distances in the formula \frac{\delta t }{\delta t}. But the formula that he uses to compute the 2 distances depends heavily on what the acceleration is like. It must be constant to use his distance formula.

But you are still using an example with a constant acceleration! .

Yes, I know. It's the exact same problem as on the test, except with different numbers. I did this to show that an example with a constant acceleration gives a different answer than a method that does take into account the gravity gradient.

Of course, because you used the wrong equation for s(t)! (you used one that describes a constant acceleration!)

I used the equation for s(t) given in the question. I changed the numbers to conform to my example, but that is the equation we were supposed to use.

...consider instead s(t) = 1 + 2 t^3...

I would not have used my method if this were the question. The fact that he says to use the function s(t)=1100-100t-16t² defines this as a constant acceleration problem, and my method works just fine in constant acceleration problems.

 

[tony873004]

1. When asked for a definition give one, don't give a description of an interpretation of something. Also, the description you gave doesn't actually make sense (for any value of x that exists?)

I just look up the word "definition" on dictionary.com and it said absolutely nothing about math.

This leads me to conclude that there is a difference between "definition" and "mathamatical definition".

So I feel I did give a definition based on the definition of the word "definition"

However, I did not give a mathamatical definition. But he did not ask for a mathamatical definition.

Had I put
<br /> f&#039;(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}<br />
as my answer, what guarantee would I have had that he wouldn't mark me wrong saying something like "That's the mathamatical definition, but I wanted you to define it in words so I can see that you understand excactly what is meant by 1st derivate. If I wanted the mathamatical definition, I would have asked for it."

What I meant by "for any value that exists" is that functions are not differentiable at certain points. If the derivative doesn't exist for a particular value of x (at a cusp, etc), then the derivative formula will not find you a derivative for that x.

 

[nrqed]

Thanks, Pat. I don't quite understand a few of your comments. I did read AKG's post.


But it does. In the teacher's method he computes average velocity by computing 2 distances and using those two distances in the formula \frac{\delta t }{\delta t}. But the formula that he uses to compute the 2 distances depends heavily on what the acceleration is like. It must be constant to use his distance formula.

You must mean *position* instead of distance. Yes, one calculates two positions, takes the difference and divide by the time interval. Tjis is the *definition* of the average velocity between two instants.
His formula does not rely on whether the acceleration is constant or not ! . For any function s(t) , the average velocity between two instants is given by his formula. I don't know why you say that the acceleration must be constant to use his formula (are thinking about the particular result he got for this problem? Of course he used the function s(t) provided! My point (and others) is that the formula v_{x average} = {s(t_f) - s(t_i) \over t_f - t_i } is valid for any function s(t), no matter if the acceleration is constant or not. On the other hand your formula works only for constant acceleration.

we all agree that your formula gives the correct answer for that particular problem (which is why you got 1.5 points!) but the issue is that, as a teacher, I would have wanted to know if you realized that your formula could only be used for the case fo a constant acceleration.

Listen, let's say you are a prof. A student uses your formula in a problem. It gives the correct answer. Is it by luck? Did the student know that it could be used because teh acceleration was constant? If it had not been constant, would the student still have used the same equation?

If the student does not mention that this special case formula v_{ave} = { v_f + v_i \over 2} can be used in that example because the acceleration is constant, how can you, as prof, tell if the student understands?

Pat

 

[tony873004]

In fact, if you don't mind my saying so, it seems weird that you used the method you used (taking the derivative and everything), it suggests that you actually have a misconception about the way average velocity should be worked out.

No, I don't mind you saying so. I want people's opinions here

The reason I used this method is because this was a test on the chapter that introduced derivatives. I could have totally pictured him saying something like "I didn't want you to use the methods you learned in physics class. I'm testing you on derivatives. I wanted you to find the answer using derivatives."

 

[AKG]

But it does. In the teacher's method he computes average velocity by computing 2 distances and using those two distances in the formula \frac{\delta t }{\delta t}. But the formula that he uses to compute the 2 distances depends heavily on what the acceleration is like. It must be constant to use his distance formula.

This only convinces me further that you were lucky to get 1.5/2. It does not depend AT ALL on what the acceleration is like. Let x(t) denote the position at time t. Then the average velocity over the interval from t₁ to t₂ is, by definition:

(x(t₂) - x(t₁))/(t₂ - t₁)

x could be ANY bizarre function. ANY! That means that acceleration can be any bizarre function as well. If you think that the formula given above only works for constant acceleration, then you've proven hands down that you don't understand the concepts, and the teacher should have the right to argue your mark down!

I would not have used my method if this were the question. The fact that he says to use the function s(t)=1100-100t-16t² defines this as a constant acceleration problem, and my method works just fine in constant acceleration problems.

Yes, but the function s(t) also "defines" it to be a problem such that the average velocity from time 1 to time 3 is whatever you computed it to be. It's all a consequence of the definitions. But you have to show that you know how to get from the definitions to the desired consequence, and you haven't shown that.

EDIT: Anyways, I think nrqed has said it best. Your solution works, but you indicated nothing to suggest that you knew what you were doing. In particular, you've done nothing on your test to show that you're not a student who thinks (v₂ - v₁)/2 works in every situation, and such a student should not get full marks, so why should you?

 

[tony873004]

His formula does not rely on whether the acceleration is constant or not ! . For any function s(t) , the average velocity between two instants is given by his formula.

I'm not sure if I'm understanding the point you're trying to make , or if you're understanding the point I'm trying to make since we seem to be saying the same things back and forth to each other.

I agree with you that his formula \frac{\delta d}{\delta t} works no matter what the acceleration is doing.

I agree with you that my formula needs constant acceleration to work.

However, his formula uses positions computed from a formula that needs constant acceleration to work.

So although \frac{\delta d}{\delta t} works for any acceleration, the inputs to that formula require constant acceleration, meaning that the process is flawed. And that's why his method and my method both get the wrong answer in my example where I threw the rock from 400,000 feet and let it drop for 1200 seconds, as the gravity gradient plays a major role in that situation.

 

[tony873004]

\frac{\delta d}{\delta t}
x could be ANY bizarre function. ANY! That means that acceleration can be any bizarre function as well.

I totally understand what you're saying. That formula works fine for any bizarre function, and mine doesn't. But the formula that generates the input for that formula (d₁and d₂ for delta d) assumes a constant acceleration.

I can not use the teacher's method to accurately compute how long it would take for an object several thousands of miles high to drop to the ground. Not because there's something wrong with \frac{\delta d}{\delta t}, but because there's something wrong with the way the inputs for that formula was generated.

 

[nrqed]

I'm not sure if I'm understanding the point you're trying to make , or if you're understanding the point I'm trying to make since we seem to be saying the same things back and forth to each other.

I agree with you that his formula \frac{\delta d}{\delta t} works no matter what the acceleration is doing.

I agree with you that my formula needs constant acceleration to work.

However, his formula uses positions computed from a formula that needs constant acceleration to work.

Good. we agree on this. But that settles the entire discussion!

So although \frac{\delta d}{\delta t} works for any acceleration, the inputs to that formula require constant acceleration, meaning that the process is flawed. And that's why his method and my method both get the wrong answer in my example where I threw the rock from 400,000 feet and let it drop for 1200 seconds, as the gravity gradient plays a major role in that situation.

This is where there is a major miscommunication. I am not sure at all what you mean by "the inputs to that formula requires constant acceleration"??! To be frank, this makes no sense to me.

What do you mean? His formula says "given a function s(t), if you want to calculate the average velocity between t_f and t_i, calculate s(t_f) and so on... You do not need to know that the acceleration is constant to perform any of these steps . Plugging a value t_f into the function s(t) requires to know the function s(t) but it does require to know that the acceleration is constant.

Obviously the prof needed to use the function s(t) to get an answer! But his way never required to know that the acceleration is constant! That's the whole point! Your method, on the other hand, requires an extra step: you must *first* check that the acceleration is constant before you can use your formula! His method does not require that extra step, it would work for any s(t). Since yoru approach requires an extra step, you should have made it clear that you did go through that extra step...just saying afterward "it's obvious that the acceleration was constant so that I could use my method" is not sufficient, from the point of view of a teacher.

Pat

 

[nrqed]

I totally understand what you're saying. That formula works fine for any bizarre function, and mine doesn't. But the formula that generates the input for that formula (d₁and d₂ for delta d) assumes a constant acceleration.

I can not use the teacher's method to accurately compute how long it would take for an object several thousands of miles high to drop to the ground. Not because there's something wrong with \frac{\delta d}{\delta t}, but because there's something wrong with the way the inputs for that formula was generated.

Now you are mixing apples and oranges. The entire discussion on how to calculate average velocities! Calculating the time to get from one point to another is a totally different issue!

 

[tony873004]

Let me try to phrase it one more way:
\frac{\delta d}{\delta t} works fine under any bizarre acceleration in this form.

But once you replace \delta d with the formulas that generate \delta d you get
<br /> \frac{1100-100(3)-16(3)^2-(1100-100(1)-16(1)^2)}{3-1}<br />

and this formula does require constant acceleration.

 

[matt grime]

However, his formula uses positions computed from a formula that needs constant acceleration to work.

But that has absolutely nothing to do with why you lost marks. And the more you assert this the more marks I would be tempted to deduct because you are displaying less and less understanding of what was going on in each post.

Secondly, if you're asked for a definition on a maths class why do you think it is acceptable to interpret that asking for a non-rigorous description? I really wouldn't try your explanation with the lecturer; he will look at you in a very funny way.

Thirdly, reread your description, it actually does not make sense as a sentence in the English language. It isn't even grammatically correct, never mind mathematically correct. It asserts something "for all x that exist" which is nonsense, the existence of the x or otherwise is immaterial, it is an existence of a limit (at x) that is important, and your sentence is not structurally capable of supporting a claim that that is what you meant.

 

[tony873004]

Now you are mixing apples and oranges. The entire discussion on how to calculate average velocities! Calculating the time to get from one point to another is a totally different issue!

You're right. I got it mixed up there. I should have said:

"I can not use the teacher's method to accurately compute how far an object will drop over a given period of time."

instead of

"I can not use the teacher's method to accurately compute how long it would take for an object several thousands of miles high to drop to the ground."

But the point I'm making is the same. s(t)=1100-100t-16t² is a constant acceleration formula and it is where delta d comes from.

 

[nrqed]

Let me try to phrase it one more way:
\frac{\delta d}{\delta t} works fine under any bizarre acceleration in this form.

But once you replace \delta d with the formulas that generate \delta d you get
<br /> \frac{1100-100(3)-16(3)^2-(1100-100(1)-16(1)^2)}{3-1}<br />

and this formula does require constant acceleration.

This is where the miscommunication lies. Why do you say this works only if the acceleration is constant? Of course, one has to use the function s(t) provided, but it does not matter at all that the acceleration is constant here!

 

[nrqed]

You're right. I got it mixed up there. I should have said:

"I can not use the teacher's method to accurately compute how far an object will drop over a given period of time."

instead of

"I can not use the teacher's method to accurately compute how long it would take for an object several thousands of miles high to drop to the ground."

But the point I'm making is the same. s(t)=1100-100t-16t² is a constant acceleration formula and it is where delta d comes from.

No, it is not where \delta d comes from! The definition v_{ave} = {\Delta x \over \Delta t} does not care on whether the acceleration is constant.

I think you are saying that the numerical value of delta d for that particular problem comes from using s(t). Of course it does! But nowhere does the fact that the acceleration is constant is required!

 

[tony873004]

...if you're asked for a definition on a maths class why do you think it is acceptable to interpret that asking for a non-rigorous description? I really wouldn't try your explanation with the lecturer; he will look at you in a very funny way...

If someone asks me for the definition of an apple, I'd reply with something like "round red fruit". I wouldn't think of giving a math formula.

Nowhere in the book does it mention "mathamatical definition". Nowhere in our lecture notes does it mention "mathamatical definition". So that's why if I'm asked for a definition I try to describe in words what it is.

He'll look at me funny anyway. We both have dry senses-of-humor and joke around with each other a lot.

 

[AKG]

You are ABSOLUTELY WRONG! There are three formulas:

v_{avg} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}

v_{avg} = \frac{s&#039;(t_2) - s&#039;(t_1)}{2}

s(t) = -16t^2 - 100t + 1100

The first is the definition of average velocity, and since it is a definition, it works for ALL s. The second is an equation for average velocity that only works when s''(t) is constant. The third is just some particular equation of the position of some hypothetical object. For that particular object, it happens that s''(t) is constant, so its acceleration is constant.

He applied formula 1 to the s that appears in formula 3. s happens to be a constant acceleration function, but whether s were constant acceleration or not makes no difference on whether he is allowed to apply 1 to 3. You applied 2 to 3, but as you've demonstrated, you don't know why it's okay to do this. The fact that applying 1 to 3 means plugging in values to formula 3, and the fact that formula 3 happens to be a constant acceleration equation has NOTHING to do with whether or not his approach was the right way to find average velocity. He did not require s to be a constant acceleration formula for his solution to work. You did, i.e. the fact that acceleration happens to be constant has EVERYTHING to do with whether or not your approach was correct, and the fact that you made no indication that you were aware of this on your tests makes his mark fair, if not overly generous.

Consider s(t) = t³. I can compute the average velocity over the interval [0, 100], it's just (100³ - 0³)/(100 - 0) = 10000. See, this method works, and it works in situations where s is not constant acceleration. Your method would give 15000, which is wrong.

Your confusing what it means for a solution to require that s be constant acceleration, and a solution to a problem where s happens to be constant acceleration. Basically, you're saying that since his solution to the problem is a solution to a problem involving a function s, where s'' is constant, his solution requires s'' to be constant. His solution to a problem involving s requires him to use the function s, and s is a constant acceleration function, so in a stupid sense, it requires him to use a constant acceleration function. Of course, the solution to any problem involving s must use s. But he doesn't use the fact that s'' is constant. You do, but you don't indicate that you're aware of this, so you were lucky to only lose 1/2 a mark.

 

[tony873004]

But nowhere does the fact that the acceleration is constant is required!

Then why does the teacher's method get the wrong answer when choosing heights and times large enough for the gravity gradient to come into play in a sifnificant manner?

 

[tony873004]

...but as you've demonstrated, you don't know why it's okay to do this...

How have I demonstrated that I don't know why it's OK to do this? I know that it's ok to do this because for scales as small as 1100 feet, gravity can be considered constant. It's the same assumption that kinmatics chapters in physics books make all the time.

 

[nrqed]

Then why does the teacher's method get the wrong answer when choosing heights and times large enough for the gravity gradient to come into play in a sifnificant manner?

I am not following you... He gets the wrong answer for what? For the average velocity? It's because you use the wrong formula for s(t)!

all we are saying is that for a given s(t), his formula will always give the correct average velocity between two times, no matter what s(t) is.

If you give an incorrect s(t) and then says his formula does not give the correct answer, it's cheating
(and notice that your formula would not give the "correct" answer either)

Pat

 

[AKG]

I totally understand what you're saying. That formula works fine for any bizarre function, and mine doesn't. But the formula that generates the input for that formula (d₁and d₂ for delta d) assumes a constant acceleration.

So what? Yes, s happens to be a constant acceleration function. But his method would have worked even if s were not constant acceleration, since his method is a simple application of the definition.

I can not use the teacher's method to accurately compute how long it would take for an object several thousands of miles high to drop to the ground. Not because there's something wrong with \frac{\delta d}{\delta t}, but because there's something wrong with the way the inputs for that formula was generated.

No, this is ABSOLUTELY WRONG! His approach would have worked just fine. His approach is just applying the definition, and that can never be wrong. It might be wrong that a function s with s'' constant can never accurately model the free fall of a real-life physical body. But his method does not involve figuring out s. The problem is not to figure out s. The problem is, given an s, find the average velocity. The problem is:

Given this s, find the average acceleration

and his method works, and will always for for any given s. If the problem were:

If you drop an object from height h with mass m, derive a formula s(t) that gives its position at time t

and you give the answer s(t) = at² + bt + c, then you've made a mistake in the physics. It wouldn't be a math mistake, it would be a physics mistake. But this is neither here nor there, since THIS WAS NOT THE PROBLEM! The problem wasn't to determine whether s(t) is the correct function to model the free fall of the object, the problem was to assume s(t) is the correct function, since that's what's given, and then compute the average velocity.

 

[AKG]

How have I demonstrated that I don't know why it's OK to do this? I know that it's ok to do this because for scales as small as 1100 feet, gravity can be considered constant. It's the same assumption that kinmatics chapters in physics books make all the time.

Again, WRONG! It's okay to apply the formula because s'' is constant. That, and only that, is the correct reason. Suppose the problem posed was this:

An object is dropped from 100ft above the ground, and it falls freely. It's position is given by s(t) = t³ + e^t - cos(t) + 1/(t+1) + 99. Find the average velocity from t=0 to t=1.

It doesn't matter that s is in enormous contradiction with physical facts. Suppose the above question were given, how would you answer it? Would you refuse to do it because it is physically incorrect? Or would you realize that, if the position is GIVEN by a function s(t), then no matter how bizarre s is, the average velocity can be computed, and using his method, it will always be correctly calculated.

 

[tony873004]

I realize what you're saying AKG, but since his formula s(t) defined this as a constant acceleration problem, I felt free to use another method that works fine for constant acceleration.

Since this was the test following the derivative chapter, it made sense to me that he would want us to solve problems using derivatives, rather than using formulas memorized from physics class.

 

[nrqed]

How have I demonstrated that I don't know why it's OK to do this? I know that it's ok to do this because for scales as small as 1100 feet, gravity can be considered constant. It's the same assumption that kinmatics chapters in physics books make all the time.

Did you explain that in your test?

we are not saying you got the wrong answer, we are saying that the way you proceeded, you had to go through an extra step to mention that you could do it that way because you knew the acceleration was constant.
If you learn a technique that works in general and then you learn a special trick that works in only some situations, and you use the special trick in a test, you should explain why it is ok to use the special trick instead of the more general method.

 

[tony873004]

Again, WRONG! It's okay to apply the formula because s'' is constant. That, and only that, is the correct reason...

We haven't covered 2nd deriatives yet, so forgive me if I don't understand this. But is s'' the rate at which acceleration changes? If so, isn't saying "It's okay to apply the formula because s" is constant" the same thing as saying "It's okay to apply the formula because gravity (acceleration) can be considered constant"?

 

[nrqed]

We haven't covered 2nd deriatives yet, so forgive me if I don't understand this. But is s'' the rate at which acceleration changes? If so, isn't saying "It's okay to apply the formula because s" is constant" the same thing as saying "It's okay to apply the formula because gravity (acceleration) can be considered constant"?

s'' = the second derivative and it represents here the rate at which the velocity changes. And yes, saying s'' is constant is saying that the acceleration is constant (you should always say "the acceleration due to gravity is constant", not "gravity is constant").

Pat

 

[tony873004]

Thanks, Pat.

I appreciate the responses of everyone who has participated in this thread. I don't spend too much time in the math forums, and I'm learning a lot from you, especially in the terminology and methods (i.e. what is a definition, state that the special condition exists, allowing me to use a shortcut, etc.)

There was another "why was I marked wrong" question I asked in post #8 in this thread. It's a bit buried now :). Does anyone have any input on that question? I though I saw a response to it last night, but now I can't find it.

 

[Hurkyl]

I realize what you're saying AKG, but since his formula s(t) defined this as a constant acceleration problem, I felt free to use another method that works fine for constant acceleration.

The problem, as has been said before, is that you didn't say why you could use that method.

As an analogy, simply guessing "16" works fine for any problem whose answer is 16. But surely you wouldn't expect to get full marks if you tried that on the test.

 

[Curious3141]

Tony, I think you're protesting too much about why you feel you're right. There has, as yet, been no real sign that you appreciate why the method you used leads one to believe that you've got your concepts wrong (and this is a bigger error to a teacher than merely getting the answer wrong).

The reason I used this method is because this was a test on the chapter that introduced derivatives. I could have totally pictured him saying something like "I didn't want you to use the methods you learned in physics class. I'm testing you on derivatives. I wanted you to find the answer using derivatives."

Now, if you'd done it the PROPER way (based on definition) and your teacher had marked you wrong and given you this (frankly, cockamamey) explanation as to why, THEN we'd all be on your side and telling you your teacher is an idiot.

As it stands, for you to have *presumed* your teacher wants you to do something in a more difficult, indirect way that doesn't apply in general cases, then complaining when you got credit deducted, makes you seem more than a little weird in your thinking.

Think about it. You're GIVEN the formula for s(t). One way involves plugging in two time values, subtracting and diving by the time. It works NO MATTER WHAT form the s(t) takes, and doesn't entail ANY assumptions about the acceleration or anything else. This is the correct, and expected way to get the answer, and it happens to be EASIER.

The way you did it involves differentiating the expression, THEN substituting time values, adding and diving by two. It ONLY works because s(t) happens to be in a particular fortuitous form. No mention was made of the fact that you're aware that what you're doing only works because of that BIG COINCIDENCE (yes, it is a big coincidence, particle mechanics questions often have inconstant accelerations). AND it is a more difficult method. One would have to question why, if you really knew what you were doing, would you bother to take more steps to get to an answer when you could've used a more correct method right from the start.

Teachers, in general and with few exceptions, are not obsessive about things. And they know a lot more than their students usually give them credit for. If you present the problem correctly worked out as it SHOULD be worked out (and you can never go wrong by starting with the definition of something), then the teacher will give you full credit whether you've covered that in your syllabus or not. If the teacher doesn't, you have a legitimate gripe and argument to get a second evaluation. But using a less reliable method simply because you think you're somehow "expected" to do that is not the way to go.

Frankly, I would've given you half a mark. Not half the credit but literally 0.5 marks out of 2. In my humble opinion, that's all your working would've deserved. I think your teacher was generous.

 

[nrqed]

Calculate the average rate of change for the indicated value of x.
f(x)=x²-5; x=3

Since we were not prohibited from using the power rule, my answer was:

f'(x)=2x
f'(3)=2*3=6

My score: 1.25/2

What the teacher wanted is:

6+h.

My reasoning as to why he's wrong:

6+h would be the average rate of change for [x, x+h], but not for x.
x is a single point. It's like asking what's the batting average of a player who has 1 hit in 1 at bat. It's 1/1. Here too, with only a single point, the denominator of a fraction designed to compute an average must contain a number n=number of data points.

So the average rate of change for a single point should be the instantaneous value of that point.

Again, thanks to everyone for playing the part of the teacher in debating against me.

the question as stated here does not make sense. To calculate an average rate of change, on must always be provided with an interval. But I have the feeling you are not really giving the full question! his answer contains "h" and you don't even mention h in the question! Are you sure he did not asked the average rate of change between x and x +h or something like that?

You cannot replace average rate of change by a derivative. These are different things. (only the limit as the interval goes to zero do they become equal)

 

[Curious3141]

Calculate the average rate of change for the indicated value of x.
f(x)=x²-5; x=3

Since we were not prohibited from using the power rule, my answer was:

f'(x)=2x
f'(3)=2*3=6

My score: 1.25/2

What the teacher wanted is:

6+h.

My reasoning as to why he's wrong:

6+h would be the average rate of change for [x, x+h], but not for x.
x is a single point. It's like asking what's the batting average of a player who has 1 hit in 1 at bat. It's 1/1. Here too, with only a single point, the denominator of a fraction designed to compute an average must contain a number n=number of data points.

So the average rate of change for a single point should be the instantaneous value of that point.

Again, thanks to everyone for playing the part of the teacher in debating against me.

In this case, if that had been the question exactly as it was posed, and your teacher had expected an answer of (6 + h), yes, your teacher is wrong. I can't even begin to see where the h would suddenly come from.

Do post the whole question, you might actually be right in this instance !