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Terminal speed

  1. Feb 27, 2005 #1
    will a golf ball of mass 45g and diamter 4.3cm reach terminal speed when dropped from a height of 25m? the drag coefficient is 0.35 and the density of air is 1.2kg/m^3.

    using the formula [tex]v_t = sqrt(\frac{2*mg}{CpA})[/tex]

    [tex]v_t = sqrt(\frac{2*(45g)(9.8m/s^2)}{(.35)(1.2kg/m^3)(0.043m)})[/tex]

    so pluggeed that into my calculator, i found the terminal speed of 220.99m/s.
    this is where i got stuck, how would i know if it reaches terminal speed or not when dropping from a height of 25m?
     
  2. jcsd
  3. Feb 28, 2005 #2
    it will not reach the terminal speed nomatter how high you drop the ball......
    the velocity of the ball is......
    v=v_t(1-e^(-kt))
    it will get closer and closer to the terminal speed when time passes, but it will never "reach" it
     
  4. Feb 28, 2005 #3
    i have a few questions, how did you get the formula v=v_t(1-e^(-kt))? and what does k and t stand for and how do i find it?
     
  5. Feb 28, 2005 #4
    solve the following DE
    [tex] -m\frac{dv}{dt}= -mg+ bv [/tex]
    b is the drag coefficient (i believe this is how ppl called it)
    k=b/m ...and t is time...
     
    Last edited: Feb 28, 2005
  6. Feb 28, 2005 #5
    sorry but a few more questions if you dont mind. how extactly would i find t? I dont think using one of the kenematics will help because i will be missing alot of info.
     
  7. Feb 28, 2005 #6
    you know calculus, do you?
    [tex] m\frac{dv}{dt}= mg- bv [/tex]
    [tex] \frac{dv}{dt}=g-\frac{b}{m}v[/tex]
    [tex] \frac{dv}{dt}=g-kv[/tex]
    [tex] dt = \frac{dv}{g-kv} [/tex]
    [tex]\int dt = \int \frac{dv}{g-kv} [/tex]
    [tex] t = \frac{-1}{k} ln(g-kv) + C [/tex]
    [tex] v(t) = \frac{g}{k}-\frac{e^{-k(t-C)}}{k} [/tex]
    [tex] v(t) = \frac{g}{k}-\frac{C'e^{-k(t)}}{k} [/tex]

    apply the initial condition [itex]v(0)=0[/itex] this implies [itex] C'=g [/itex]
    therefore,
    [tex] v(t) = \frac{g}{k}-\frac{ge^{-k(t)}}{k} [/tex]
    [tex] v(t) = \frac{g}{k}(1-e^{-k(t)}) [/tex]
    [tex] v(t) = \frac{gm}{b}(1-e^{-mt/b}) [/tex]

    whereas gm/b is the terminal velocity
     
  8. Feb 28, 2005 #7
    ah i see, thanks for the help!
     
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