Modelling Question
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Dr.Peterson
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I'm wondering if the book from which this comes has discussed any modeling issues that might be relevant -- that is, is this question testing something they have taught, rather than just your intuition? If I were tutoring you in person, I would be looking through this chapter to see what they have said.
The main possibility I see is that a bounce takes a non-zero time to occur, so the "time of a bounce" is not quite well-defined; but if the model is appropriately used to predict height, then it seems appropriate to use it to predict when the height is zero, allowing for some error. I see no more error likely in the times (which are very simple) than in the heights.
If this were a physics text, and explained why the model is used, there might be more to say. It seems to me that two factors would combine to reduce the height of each bounce, namely air resistance (which is a continuous effect) and friction during the bounce (which occurs discretely, and can't be modeled well by the exponential factor, which is continuous). So the model would seem to be paying more attention to the less important effect, and therefore might be bad for any purpose.
The main possibility I see is that a bounce takes a non-zero time to occur, so the "time of a bounce" is not quite well-defined; but if the model is appropriately used to predict height, then it seems appropriate to use it to predict when the height is zero, allowing for some error. I see no more error likely in the times (which are very simple) than in the heights.
If this were a physics text, and explained why the model is used, there might be more to say. It seems to me that two factors would combine to reduce the height of each bounce, namely air resistance (which is a continuous effect) and friction during the bounce (which occurs discretely, and can't be modeled well by the exponential factor, which is continuous). So the model would seem to be paying more attention to the less important effect, and therefore might be bad for any purpose.
It was an exam question from an A level maths paper (UK) which was sat yesterday.
To me it's ambiguous, it could just ask you to comment on how suitable the model is? In which case i would probably question the equal time intervals between each bounce and maybe that's what the examiner is looking for.
But the way the question is phrased 'explain why this model should not be used to predict the time..' to me suggested that it's less suitable to predict time intervals than height??? And got me thinking along the lines of dependent and independent variables but i don't think that's the answer?? It feels like one of those ' guess what's in my head' questions.
The mark scheme will come out in a few months and i will post what they say! Thanks anyway.
To me it's ambiguous, it could just ask you to comment on how suitable the model is? In which case i would probably question the equal time intervals between each bounce and maybe that's what the examiner is looking for.
But the way the question is phrased 'explain why this model should not be used to predict the time..' to me suggested that it's less suitable to predict time intervals than height??? And got me thinking along the lines of dependent and independent variables but i don't think that's the answer?? It feels like one of those ' guess what's in my head' questions.
The mark scheme will come out in a few months and i will post what they say! Thanks anyway.
Dr.Peterson
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I agree with you; but you haven't answered my question as to what topics are covered on the exam, and what has been taught about it. (I have no knowledge of "A level maths papers", or how standardized the curriculum is, but I would expect there to be some indication of what you are expected to know, and that questions would be based on some such standard.)
Mathematically modelling is called an overarching theme in the A level papers and in short this is described in the specification:
Does this answer your question or do you mean specific mathematical topics?
Dr.Peterson
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It may well be that the answer to my question is hidden in curriculum material rather than displayed clearly in standards; I've never found standards helpful for answering specific questions.
Clearly OT3.4 is the reason they ask this question; but it doesn't elaborate on what kinds of reasons for appropriateness are taught and tested. It mentions "simplifying assumptions", but nothing was said here about the making of the model, much less what assumptions were made (though that is where my thoughts went when I made comments on the physics).
If I really wanted to figure out what the question wants, I'd still want to read your textbook or whatever other specific materials there are. (For instance, if differential equations are part of the curriculum, perhaps you would be expected to recognize how this model would have been developed, and hence what assumptions were made.) But probably, the only thing to do is to wait -- or perhaps to look at the mark schemes for past exams, since those likely have questions about appropriateness of models, too. I'd be interested in seeing similar questions and their answers.
Clearly OT3.4 is the reason they ask this question; but it doesn't elaborate on what kinds of reasons for appropriateness are taught and tested. It mentions "simplifying assumptions", but nothing was said here about the making of the model, much less what assumptions were made (though that is where my thoughts went when I made comments on the physics).
If I really wanted to figure out what the question wants, I'd still want to read your textbook or whatever other specific materials there are. (For instance, if differential equations are part of the curriculum, perhaps you would be expected to recognize how this model would have been developed, and hence what assumptions were made.) But probably, the only thing to do is to wait -- or perhaps to look at the mark schemes for past exams, since those likely have questions about appropriateness of models, too. I'd be interested in seeing similar questions and their answers.