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Math Help - Need help with this velocity question

  1. #1
    Junior Member
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    Need help with this velocity question

    Hi can someone help me figure this out:

    position of a particle (s) at time t:
    s = 2t^3 + 18t^2 - 672t

    I found the total distance traveled during the first 19 secs is 7448 (right?)

    I need help figuring this out:

    the particle is at rest after ___ seconds. <- is it 8? the answer doesn't work

    also, The particle is moving forward in the time interval t>8 and backward in the time interval t<8, is this right?

    thanks!
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  2. #2
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    Hello, coldfire!

    I will assume that: t \geq 0


    Position (s) of a particle at time t: .  s(t) \:= \:2t^3 + 18t^2 - 672t

    I found the total distance traveled during the first 19 secs is 7448, right? . . . . no


    I need help figuring this out:
    The particle is at rest after ___ seconds.
    . . Is it 8? . . .the answer doesn't work . . . . yes, it does

    Also, the particle is moving forward in the time interval t > 8
    and backward in the time interval t < 8, is this right? . . . . Yes!

    The position function is: . s(t) \;=\;2t^3 + 18t^2 - 672t

    The velocity function is: . v(t) \:=\:s'(t) \:=\:6t^2 + 36t - 672

    The velocity is zero when: . 6t^2 + 36t - 672 \:=\:0 \quad\Rightarrow\quad t^2 + 6t - 112 \:=\:0

    . . Factor:. (t-8)(t+14) \:=\:0 \quad\Rightarrow\quad t \;=\;8,\:{\color{red}\rlap{/////}}-14


    Let's examine the behavior of the particle.


    At t = 0:\;\;s(0) = 0,\;v(0) = -672
    . . It is at the origin and is moving to the left.

    At t = 8:\;\;s(8) = -3200,\;v(8) = 0
    . . It has moved 3200 units to the left ... and has stopped.
    An instant later, it is moving to the right.

    At t=19:\;s(19) = +7448,\;v(19) = 2178
    . . It has moved: . 7448 - (-3200) \:=\:10,648 units to the right.


    It moved a total of: . 32,000 + 10,648 \;=\;\boxed{13,048\text{ units}}

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  3. #3
    Junior Member
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    ohh I get it now!

    that was very helpful, thank you!

    (answer was 13,848 by the way, I saw where you made a small mistake in the addition)

    So it's basically the same thing as saying:

    |s(8)-s(0)| + |s(19)-s(8)|?
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