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How to get the distance traveled from the force and mass functions?

  1. Sep 27, 2012 #1
    1. The problem statement, all variables and given/known data
    I have two functions:
    F(t) - where F(t) is the force at a given time t
    m(t) - where m(t) is the mass of the object in question at a given time t

    Let's say that some force (in a thrust form) is applied to the object for "b" seconds. The function F(t) specifies in what manner.

    How can I get the distance traveled by the object after "b" seconds, if we know that the velocity, acceleration and distance traveled are all 0 at t = 0?


    2. Relevant equations
    -


    3. The attempt at a solution
    I've tried using an analogy of the Riemann sum (diving each instantaneous force by each instantaneous mass and summing everything for an acceleration-time function), and it turned out to be too tedious and imprecise to be applied practically.
     
  2. jcsd
  3. Sep 28, 2012 #2
    Anyone?
     
  4. Sep 28, 2012 #3

    gneill

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    Staff: Mentor

    You can write an expression for the acceleration with respect to time from the given force and mass functions. Integrate to find velocity. Integrate again to find distance.
     
  5. Sep 28, 2012 #4
    Thanks for your feedback. Can one just successively integrate the force function dived by the mass function to get the distance function?
     
  6. Sep 28, 2012 #5

    gneill

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    Sure.
     
  7. Sep 28, 2012 #6

    HallsofIvy

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    Staff Emeritus
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    m(t)dv/dt= f(t) so that dv= (f(t)/m(t))dt and you can integrate that. Once you have found v(t), you can use dx/dt= v(t) and integrate dx= v(t)dt to find the distance function. Of course, there is no guarentee that any of those functions will be "integrable" as an elementary function.
     
  8. Sep 29, 2012 #7
    But what if the F(t) and m(t) functions aren't continuous, and are only continuous on the interval of [0, b]? It's easy to do the calculations when the function is completely continuous. But how to do it if it's continuous only over [0, b], and we want to know the distance traveled at b?
     
    Last edited: Sep 29, 2012
  9. Sep 29, 2012 #8

    gneill

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    :confused: If they're continuous over [0,b] and you want the distance at b, I don't see the problem since the integrals will be defined over the domain.

    If the functions are not continuous then it is up to you to interpret their behavior in terms of physical laws and deal with the implications. This might, for example, mean splitting the domain of integration into continuous pieces and "bridging" the gaps with assumed constant velocity sections.
     
  10. Sep 29, 2012 #9
    Can one use definite integration for that?
     
  11. Sep 29, 2012 #10

    gneill

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    Sure. Any integration is just a sum. A sum can be split into chunks and added separately. If some physics occurs between the parts represented by the integrations, then the integrations just become terms in an overall equation of motion where you stick other terms to fill in the "spaces".

    Do you have some particular F(t) and M(t) in mind which is raising these concerns?
     
  12. Sep 29, 2012 #11
     
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