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Math Help - Velocity of particle using graph? Physics?

  1. #1
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    Velocity of particle using graph? Physics?

    I'm not sure how to figure this out:Velocity of particle using graph? Physics?-capture10.jpgI'm not sure how to find how far the particle with using this graph? Any help? I took physics two years ago before college and I know I learned about this but I can't remember what to do, using either physics or calc methods. Thanks!
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  2. #2
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    Re: Velocity of particle using graph? Physics?

    $a(t) = 3t$

    Acceleration is the derivative of velocity: $a(t) = v'(t)$. So:

    $\displaystyle v(t) = \int a(t) dt = \dfrac{3}{2}t^2+C$

    You are told initial velocity is 5. Initial velocity means t=0. So:

    $v(0) = 5 = \dfrac{3}{2}(0)^2 + C$

    shows $C=5$.

    Hence, $v(t) = \dfrac{3}{2}t^2+5$.

    Velocity is the derivative of position: $v(t) = s'(t)$. So, $\displaystyle s(t) = \int v(t)dt$.

    Now, you are looking for the distance the particle has moved. That is just $\displaystyle s(3)-s(0) = \int_0^3 v(t)dt$:

    $\displaystyle \int_0^3 v(t)dt = \int_0^3 \left(\dfrac{3}{2}t^2+5\right)dt = \left[\dfrac{1}{2}t^3 + 5t\right]_0^3 = 19.5$
    Thanks from canyouhelp
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  3. #3
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    Re: Velocity of particle using graph? Physics?

    Quote Originally Posted by SlipEternal View Post
    $a(t) = 3t$

    Acceleration is the derivative of velocity: $a(t) = v'(t)$. So:

    $\displaystyle v(t) = \int a(t) dt = \dfrac{3}{2}t^2+C$

    You are told initial velocity is 5. Initial velocity means t=0. So:

    $v(0) = 5 = \dfrac{3}{2}(0)^2 + C$

    shows $C=5$.

    Hence, $v(t) = \dfrac{3}{2}t^2+5$.

    Velocity is the derivative of position: $v(t) = s'(t)$. So, $\displaystyle s(t) = \int v(t)dt$.

    Now, you are looking for the distance the particle has moved. That is just $\displaystyle s(3)-s(0) = \int_0^3 v(t)dt$:

    $\displaystyle \int_0^3 v(t)dt = \int_0^3 \left(\dfrac{3}{2}t^2+5\right)dt = \left[\dfrac{1}{2}t^3 + 5t\right]_0^3 = 19.5$
    Thanks for showing the steps, I couldn't put that together because I was so focused on it being a physics problem I didn't even realize I could apply these steps.
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  4. #4
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    Re: Velocity of particle using graph? Physics?

    Physics involves a lot of applied math.
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