Various position functions from dx/dt

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Discussion Overview

The discussion revolves around the relationship between velocity and position functions in calculus, particularly how different forms of position functions can arise from a given velocity function. Participants explore the implications of constant versus variable acceleration and the derivation of one function from the other.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question why a specific form of the position function (s(t) = 1/2at^2 + v0t + v0) is expected from the velocity function v(t) = dx/dt, suggesting that multiple forms of position functions can exist.
  • Others express confusion about how to derive v(t) from s(t) without knowing the function s(t), raising concerns about how motion problems can be solved with only known variables and parameters.
  • One participant mentions that certain functions are reasonable models for projectile motion, indicating that more complex functions may account for factors like air resistance.
  • There is a discussion about the dimensional consistency of velocity functions, with participants questioning how terms like t^2 can appear in velocity equations without clear constants, and how this relates to the units of measurement.
  • Some participants clarify that coefficients in velocity functions must have specific dimensions to maintain consistency with the units of distance and time.

Areas of Agreement / Disagreement

Participants express differing views on the derivation of position and velocity functions, with no consensus on the necessity of specific forms or the implications of dimensional analysis. The discussion remains unresolved regarding the conditions under which different position functions can arise from a given velocity function.

Contextual Notes

Limitations include the lack of clarity on the assumptions regarding acceleration (constant vs. variable) and the specific forms of functions used in motion problems. The discussion also highlights the dependence on experimental data to determine functions in practical scenarios.

thedy
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Hi,I am just starting to solve calculus problems but,how is it possible,that from v(t) we can get so many differently position function.I mean,if we have v(t)=dx/dt we should always get just v=at+v0,and from this to get s(t)=1/2at^2+v0t+v0.But I have found many motion problems in which position functions have different forms,like for example:s(t)=t^2+4t^2 or s(t)=-t-t^2.And many others like if acceleration is not constant and so on.

So my question.If we solve some problem,we always start with v=dx/dt and therefore,we should get always just one,and still the same form of position function,should not we?

thanks for answers
 
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thedy said:
I mean,if we have v(t)=dx/dt we should always get just v=at+v0
thanks for answers
Why should we always get that?
 
I will ask otherwise.How do I get v(t) by derivation of s(t) if I do not know function s(t)?If I have some motion problem where we have function s(t) like known stuff,from what is this function determined?I mean,how I can know s(t) if I do not know v(t).And how I could know v(t) if I do not know s(t)?
If we are trying to solve some motion problem and we have just known variables and parameters,but we have no function,how we can solve the problem?I mean,it is determined,this function,which is neccessery only experimentally?
Thanks
 
Last edited:
arildno said:
Why should we always get that?
OK,well when we get other form?Not at+v0?If we still do dx/dy?I know,we must know x.What is hidden in ,,x".And that is my question.How we know,or where or from we can find out what is ,,x"?
Thanks
 
thedy said:
I will ask otherwise.How do I get v(t) by derivation of s(t) if I do not know function s(t)?
If you don't know the formula for s(t), you can still get v(t) (or at least an approximation of it) by using numerical techniques of differentiation. For example, if you have these data points - (0, 2), (1, 5), (2, 10) - with the first coordinate being the time t and the second coordinate being position s, you can choose a polynomial or other function to fit the data to, and then differentiate. Alternatively, you can find the slope between successive pairs of points to get approximate values for v(t).
thedy said:
If I have some motion problem where we have function s(t) like known stuff,from what is this function determined?
Certain functions are reasonable models for projectile motion. The simpler functions such as s(t) = 16t2 give OK results and are easy to work with, but only take into account the gravitational force. For better results that take air resistance into account, the functions are more complicated, but give results that are more true to life. In general the equations of motion are arrived at by experimentation over hundreds of years (going back to Galileo).
thedy said:
I mean,how I can know s(t) if I do not know v(t).And how I could know v(t) if I do not know s(t)?
If we are trying to solve some motion problem and we have just known variables and parameters,but we have no function,how we can solve the problem?I mean,it is determined,this function,which is neccessery only experimentally?
Thanks

thedy said:
OK,well when we get other form?Not at+v0?If we still do dx/dy?I know,we must know x.What is hidden in ,,x".And that is my question.How we know,or where or from we can find out what is ,,x"?
Thanks
 
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Thank you very much for answers. :)
 
Hi,I just want to ensure.If I have for example velocity function:v(t)=2t-t^2,then t^2 expression is only time over 2 or some constant is next to t^2,but this constant=1?Or how is it possible,that we have only t^2,it seems,like we could have function that for example is s(t)=t^2 and then meter=second? Dimensionally seems to be incorrect...
THanks
 
thedy said:
Hi,I just want to ensure.If I have for example velocity function:v(t)=2t-t^2,then t^2 expression is only time over 2 or some constant is next to t^2,but this constant=1?Or how is it possible,that we have only t^2,it seems,like we could have function that for example is s(t)=t^2 and then meter=second? Dimensionally seems to be incorrect...
THanks
The constants wouldn't be dimensionless. In your formula, v(t) = 2t - t2, the coefficient of t would have to be <distance>/<time2> and the coefficient of t2 would have to be <distance>/<time3>. That way the right side would be in units of <distance>/<time> to match the left side.

On the other hand, maybe the problem isn't concerned with units, in which case all they're interested in is the numeric value of the velocity.
 
Mark44 said:
The constants wouldn't be dimensionless. In your formula, v(t) = 2t - t2, the coefficient of t would have to be <distance>/<time2> and the coefficient of t2 would have to be <distance>/<time3>. That way the right side would be in units of <distance>/<time> to match the left side.

On the other hand, maybe the problem isn't concerned with units, in which case all they're interested in is the numeric value of the velocity.
Yes,but what is the value of coefficient?For example the value of the coeffcient of t2 is 1?
 
Last edited:
  • #10
thedy said:
Yes,but what is the value of coefficient?For example the value of the coeffcient of t2 is 1?
Yes. If the coefficient of some variable is 1, we usually don't bother to write it. For example x and 1x mean the same thing.
 

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