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Thread: Differential equation

  1. #1
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    Differential equation

    Hi,
    I am kinda having trouble with the following problem:

    Suppose that the acceleration of a model rocket is proportional to the difference between 120 ft/sec and the rocket's velocity. If it is initially at rest and its initial acceleration is 120 ft/sec^2, how long will it take to accelerate to 96 ft/s?
    t = ______________ sec


    What I have tried:

    dv/dt=k(120-v)

    integrating both sides:

    -ln|120-v| = t + C

    Since, t = 0 and initial velocity v = 0

    Thus, C = -ln|120|

    I am stuck here:

    Please help,

    Many thanks....
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  2. #2
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    Quote Originally Posted by althaemenes View Post
    Hi,
    I am kinda having trouble with the following problem:

    Suppose that the acceleration of a model rocket is proportional to the difference between 120 ft/sec and the rocket's velocity. If it is initially at rest and its initial acceleration is 120 ft/sec^2, how long will it take to accelerate to 96 ft/s?
    t = ______________ sec

    $\displaystyle a = k(120-v)$

    $\displaystyle a(0) = 120$ , $\displaystyle v(0) = 0$

    $\displaystyle 120 = k(120)$ ... $\displaystyle k = 1$


    $\displaystyle \frac{dv}{dt} = 120-v$

    $\displaystyle \frac{-dv}{120-v} = -dt$

    $\displaystyle \ln|120-v| = -t + C$

    $\displaystyle 120-v = Ae^{-t}$

    $\displaystyle v = 120 + Ae^{-t}$

    $\displaystyle v(0) = 0$

    $\displaystyle 0 = 120 + A$ ... $\displaystyle A = -120$

    $\displaystyle v = 120(1 - e^{-t})
    $

    $\displaystyle a = 120e^{-t}$

    $\displaystyle 96 = 120e^{-t}$

    solve for t
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  3. #3
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    Ans not right!!!

    Hi, for t I got 0.22314 or ln(120/96) but its not right....


    Please help,

    I have 1 trial left.
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  4. #4
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    ... how long will it take to accelerate to 96 ft/s?
    I misread the statement ... 96 ft/s is a velocity

    $\displaystyle 96 = 120(1-e^{-t})$

    now solve for t
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