? asked in Science & MathematicsMathematics · 1 decade ago

calculus problem velocity?

The velocity v of an object falling through the earth's atmosphere obeys the DE

dv/dt=g-kv

where k is called the drag constant and g is the gravitional constant. This equation states that the acceleration of the object is g reduced by an amount that represents air resistance(kv). Air resistance is proportional to the velocity of the object.

A find the function v(t) assuming that v(0)=0

B Let vT=g/k. When v=vT, the acceleration dv/dt is zero and thus the object maintains a constant terminal velocity. Rewirte v(t) in terms of vT and sketch the graph of v(t).

Update:

C For k=.2 how long does it take for an object to reach 95% of its terminal velocity?

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  • 1 decade ago
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    A)

    dv/dt = g - kv

    dv/(g - kv) = dt

    dv/(g/k - v) = kdt

    ln(g/k - v) = -kt + ln(C)

    g/k - v = Ce^(-kt)

    at v(0) = 0

    g/k - 0 = Ce^(-k*0)

    C = g/k

    g/k - v = g/k*e^(-kt)

    v(t) = g/k*(1 - e^(-kt))

    B)

    v(t) = vT*(1 - e^(-kt))

    C)

    v/vT = 1 - e^(-kt)

    0.95 = 1 - e^(-0.2t)

    e^(-0.2t) = 1 - 0.95

    -0.2t = ln(0.05)

    t = -ln(0.05)/0.2

    t = 15.0 s

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