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Math Help - Question about integration + motion

  1. #1
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    Question about integration + motion

    20) A particle moves in a straight line . At time t seconds its displacement is x metres from a fixed point O on the line, its acceleration is a ms^-2, and its velocity is v ms^-1 where v is given by v=32/x - x/2.

    a) find an expression for a in terms of x.

    b) Show that t = integrate(2x / (64-x^2)), and hence show that x^2 = 64 - 60 e^-t.


    I've done question a,

    a)
    v^2 = 1024/x^2 - 2(32/x * x/2) + x^2/4
    v^2 = 1024/x^2 + x^2/4 -32
    1/2 v^2 = 512/x^2 + x^2/8 -16
    d/dx 1/2 v^2 = -1024/x^3 + x/4
    a= x/4 -1024/x^3

    b)
    v = dx/dt = (64-x^2)/2x
    dt/dx = 2x/(64-x^2)
    t = integrate (2x/(64-x^2))
    t= -ln(64-x^2) +C
    C-t = ln(64-x^2)
    e^(C-t) = 64-x^2
    x^2 = 64 -e^(C-t)

    then what should I do to find out the C
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  2. #2
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    Quote Originally Posted by hkdrmark View Post
    20) A particle moves in a straight line . At time t seconds its displacement is x metres from a fixed point O on the line, its acceleration is a ms^-2, and its velocity is v ms^-1 where v is given by v=32/x - x/2.

    a) find an expression for a in terms of x.

    b) Show that t = integrate(2x / (64-x^2)), and hence show that x^2 = 64 - 60 e^-t.


    I've done question a,

    a)
    v^2 = 1024/x^2 - 2(32/x * x/2) + x^2/4
    v^2 = 1024/x^2 + x^2/4 -32
    1/2 v^2 = 512/x^2 + x^2/8 -16
    d/dx 1/2 v^2 = -1024/x^3 + x/4
    a= x/4 -1024/x^3

    b)
    v = dx/dt = (64-x^2)/2x
    dt/dx = 2x/(64-x^2)
    t = integrate (2x/(64-x^2))
    t= -ln(64-x^2) +C
    C-t = ln(64-x^2)
    e^(C-t) = 64-x^2
    x^2 = 64 -e^(C-t)

    then what should I do to find out the C
    x^2 = 64 - e^{C-t} = 64 - e^{C} e^{-t} = 64 - A e^{-t}.

    To get A you need a boundary condition that gives the value of x for a value of t. An obvious one to consider is x = 0 when t = 0, but v (and a) is undefined at x = 0 so that's no good .....

    The answer you have to get seems to assume the boundary condition x = 2 when t = 0. From the information given there's no good reason where this boundary condition has come from ....
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