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Math Help - Integration word problem please help

  1. #1
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    Integration word problem please help

    Lunar Gravity

    On the moon, the acceleration due to gravity is -1.6 m/(sec^2). A stone is dropped from a cliff on the moon and hits the surface of the moon 20 seconds later. How far did it fall? What was its velocity at impact?




    So far i have s(20)=0
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  2. #2
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    Because acceleration is defined as the rate of increase of velocity, and velocity the rate of increase of distance, we know that

    \frac{d}{dt}\frac{d}{dt}s(t)=\frac{d^2s}{dt^2}=-1.6.

    To find velocity from acceleration, we integrate:

    \frac{ds}{dt}=\int \frac{d^2 s}{dt^2}\,dt=-1.6t+C.

    As the stone is dropped rather than thrown, its velocity at t=0 must be 0, and therefore C must be 0:

    \frac{ds}{dt}=-1.6t.

    Now all that remains is to perform the second integration

    s(t)=\int \frac{ds}{dt}\,dt=\int -1.6t\,dt,

    and find the value of C for which s(20)=0.
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