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Math Help - Integral

  1. #1
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    Integral

    The position of to particles on the x-axis are given

    S1 = 3t^3 - 12t^2 + 18t + 5 and
    S2 = -3^3 + 9t^2 - 12t , where "t" is measured in seconds.

    When do the particles have the same speed?

    I cannot figure out how to derivate/integrate these.. Someone help?
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  2. #2
    Senior Member DivideBy0's Avatar
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    You can use the power rule to differentiate the expression:

    \frac{d}{dx} (x^n)=nx^{n-1}

    The derivatives of S_1 and S_2 are:

    S_1\ '(t)=9t^2-24t+18

    S_2\ '(t)=-9t^2+18t-12

    And since they represent the instantaneous velocity of the particle, you are looking for when S_1\ '(t)=S_2\ '(t)

    Solving for t, I get complex answers... so that would mean the particles never travel at the same speed.

    In fact, knowing that the gradient of a function is its rate of change, you can graph and observe the original two functions S_1(t) and S_2(t) and notice that they are cubics which are a reflection of one another. In S_1, the gradient is always positive. In S_2, the gradient is always negative. This shows that their gradients and hence their rates of change can never be equal.
    Last edited by DivideBy0; October 7th 2007 at 01:20 PM.
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  3. #3
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    S2 = -t^3 - 24t + 18

    my bad.

    But how do i figure their speed from this?
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  4. #4
    Senior Member DivideBy0's Avatar
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    In that case, applying the power rule, you get

    \frac{d}{dx}(-t^3-24t+18)=-3t^2-24

    Equating S_1\ '(t) and S_2 \ '(t):

    -3t^2-24=9t^2-24t+18

    Once again I get complex numbers for t, meaning the particles don't ever have the same speed, but I'm in a bit of a hurry at the moment, so you'll have to check over that.
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