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Solving for T in a horizontal projectile equation

  1. Apr 12, 2003 #1
    Hi, I am stuck on even how to start to solve for T

    the equations is:

    0 = Vi / K (1 - e^-kt)(sin theta) + (g / k^2)(1-kt-e^-kt)

    Any suggestions on how to begin to solve for T would be appreciated.

    Thanks,

    Matt
     
  2. jcsd
  3. Apr 12, 2003 #2

    FZ+

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    My oh my is this complicated....

    Hmm... your problem is this bit:

    1-kt-e^-kt

    Since you have t both inside and outside the e^, I don't think there is an algebraic method for you to get an exact answer. Are you sure you formed the equation correctly?

    I may be wrong though...
     
  4. Apr 12, 2003 #3
    Yes, it is correct :(
     
  5. Apr 13, 2003 #4

    HallsofIvy

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    HallsofIvy

    Then you will need either to use a numerical form of solution or do a google search form "Lambert W function".
     
  6. Apr 13, 2003 #5

    enigma

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    Couldn't you do it iteratively?

    Pick a value for T and solve. Then take that result and plug it in for T and repeat.

    If all goes well (depending on your pick to start...), it will converge on an answer.
     
  7. Apr 14, 2003 #6

    HallsofIvy

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    Pick a value of T and solve for what? :smile: What you are describing (I think!) is one very crude way of solving an equation iteratively.

    Newton's method will work faster.
     
  8. Apr 14, 2003 #7
    Sorry if this seems dumb ...
    But how did the equation of an horizontal projectil get this complicated ?
    The horizontal projectile equations that i know are well too easier ! (The one derived from the SUVAT equations)
     
  9. Apr 14, 2003 #8
    The equation is for air drag, notice the k.
     
  10. Apr 14, 2003 #9
    Ok, I got it down to

    10^1+.5t = 19.6t

    Can anyone help me on the Omega function?
     
  11. Apr 15, 2003 #10
    This is what I did:

    I inserted some fixed constants and multiplied out

    (48/.5)(1-^e-.5t)(sin 45) + (9.8/.25)(1-(.5t) - e^-.5t) = 0

    (96 - 96e^-.5t)(sin 45) + 39.2(1-(.5t)-e^-.5t) = 0

    67.88 - 67.88^e-.5t + 39.2 - 19.6t - 19.6e^-.5 = 0

    107.2 - 87.48e^-.5t - 19.6t = 0

    107.2 - 19.6t = 87.48^e-.5t

    log(107.2 - 19.6t) = log(87.48^e-.5t)

    log107.2 - log19.6t = -.5tLog(87.48)

    2.030194 - log19.6t = -.5t(1.94198)

    1.045463 - log19.6t = -.5t

    -(1.045463 - log19.6t = -.5t)

    -1.045463 + log19.6t = .5t

    log19.6t = .5t + 1.045463

    10^(.5t + 1.045463) = 19.6t

    This is where Im stuck.
     
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