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Thread: Acceleration

  1. #1
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    Exclamation Acceleration

    A plane lands on a runway at 100m/s. It then brakes with a constant deceleration until it stops 2 km down the runway. Acceleration = -k, for some positive constant k. By integrating with respect to x, find k, and find v^2 as a function of x.

    Could someone please help?
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  2. #2
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    Quote Originally Posted by Sunyata View Post
    A plane lands on a runway at 100m/s. It then brakes with a constant deceleration until it stops 2 km down the runway. Acceleration = -k, for some positive constant k. By integrating with respect to x, find k, and find v^2 as a function of x.

    Could someone please help?
    $\displaystyle
    \frac{dv}{dt} = -k
    $

    $\displaystyle
    v = \frac{dx}{dt}
    $

    $\displaystyle
    \frac{dv}{dt} \cdot \frac{dt}{dx} = -\frac{k}{v}
    $

    $\displaystyle \frac{dv}{dx} = -\frac{k}{v}$

    $\displaystyle
    v \, dv = -k \, dx
    $

    $\displaystyle \frac{v^2}{2} = -kx + C$

    $\displaystyle
    v^2 = -k_2 x + C_2
    $

    at $\displaystyle x = 0$ , $\displaystyle v = 100$ m/s

    $\displaystyle
    100^2 = C_2
    $

    $\displaystyle v^2 = -k_2 x + 100^2$

    at $\displaystyle x = 2000$ m , $\displaystyle v = 0$

    $\displaystyle 0 = -k_2(2000) + 100^2$

    $\displaystyle
    k_2 = 5
    $

    $\displaystyle
    v^2 = -5x + 100^2
    $
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