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Math Help - an AP Calculus AB review question

  1. #1
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    an AP Calculus AB review question

    A body is coasting to a stop and the only force acting on it is a resistance proportional to its speed, according to the equation ds/dt = v[final] = v[original] e^-(k/m)t; s(0) = 0, where v[original] is the body's initial velocity (in m/s), v[final] is its final velocity, m is its mass, k is constant, and t is time.

    v[final]=1;v[original]=30;m=50;k=1.5;t=113.383

    b) How far, to the nearest 10 meters, will the body coast during the time it takes to slow from 30m/s to 1m/s?


    I found the first part which gave me my time, but I don't know how to continue from here. If anyone has any advice to offer, I'd greatly appreciate it. Thanks.
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  2. #2
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    Quote Originally Posted by Jziffra View Post
    A body is coasting to a stop and the only force acting on it is a resistance proportional to its speed, according to the equation ds/dt = v[final] = v[original] e^-(k/m)t; s(0) = 0, where v[original] is the body's initial velocity (in m/s), v[final] is its final velocity, m is its mass, k is constant, and t is time.

    v[final]=1;v[original]=30;m=50;k=1.5;t=113.383

    b) How far, to the nearest 10 meters, will the body coast during the time it takes to slow from 30m/s to 1m/s?


    I found the first part which gave me my time, but I don't know how to continue from here. If anyone has any advice to offer, I'd greatly appreciate it. Thanks.
    \Delta x = \int_{t_1}^{t_2} v(t) \, dt<br />
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    Thank you.

    I've run into a problem with the third part in that I'm getting an illogical answer for my time:

    c) If the body coasts from 30m/s to a stop, how far will it coast?

    it's given that ds/dt = v[final] = v[original]e^-(k/m)t; I know that m = 50, k = 1.5, v[original] = 30 and I'm trying to find t when v[final] = 0, but that gives me ln(0/30)/(-1.5/50) which is impossible with the natural log of 0.

    Does anybody know where I went wrong here?
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  4. #4
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    Quote Originally Posted by Jziffra View Post
    Thank you.

    I've run into a problem with the third part in that I'm getting an illogical answer for my time:

    c) If the body coasts from 30m/s to a stop, how far will it coast?

    it's given that ds/dt = v[final] = v[original]e^-(k/m)t; I know that m = 50, k = 1.5, v[original] = 30 and I'm trying to find t when v[final] = 0, but that gives me ln(0/30)/(-1.5/50) which is impossible with the natural log of 0.

    Does anybody know where I went wrong here?
    v(t) \to 0 as t \to \infty

     <br />
\Delta x = \lim_{b \to \infty} \int_0^b v(t) \, dt<br />
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