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Thread: The Derivative as a Rate of Change

  1. #1
    Member VitaX's Avatar
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    The Derivative as a Rate of Change

    A rock thrown vertically upward from the surface of the moon at a velocity of $\displaystyle 24$ $\displaystyle \frac{m}{sec}$ reaches a height of $\displaystyle s=24t-0.8t^2$ meters in $\displaystyle t$ $\displaystyle sec$.
    a. Find the rock's velocity and acceleration at time $\displaystyle t$ $\displaystyle \rightarrow v=24-1.6t$ $\displaystyle \frac{m}{s}$ $\displaystyle a=-1.6$ $\displaystyle \frac{m}{s^2}$
    b. How long does it take the rock to reach its highest point? $\displaystyle \rightarrow 24-1.6t=0$ $\displaystyle t=15$ $\displaystyle sec$
    c. How high does the rock go? $\displaystyle \rightarrow s_{max}=24(15)-0.8(15)^2$ $\displaystyle s_{max}=180$ $\displaystyle m$
    d. How long does it take the rock to reach half its maximum height?
    e. How long is the rock aloft? $\displaystyle \rightarrow 30$ $\displaystyle sec$

    I don't know how to find how long it takes the rock to reach half its max height. How would I solve this?
    Last edited by VitaX; Oct 10th 2009 at 08:49 PM.
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  2. #2
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by VitaX View Post
    I don't know how to find how long it takes the rock to reach half its max height. How would I solve this?
    Well, once ya find the max height (call it $\displaystyle s_{max}$) then throw $\displaystyle \frac{s_{max}}{2}$ into the left side of your displacement function and then solve for $\displaystyle t$.

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  3. #3
    Member VitaX's Avatar
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    Quote Originally Posted by VonNemo19 View Post
    Well, once ya find the max height (call it $\displaystyle s_{max}$) then throw $\displaystyle \frac{s_{max}}{2}$ into the left side of your displacement function and then solve for $\displaystyle t$.

    overlooked that easy bit. So when I put them in the quadratic formula, do I disregard one answer or take both as when its going up and then going down?
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  4. #4
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by VitaX View Post
    overlooked that easy bit. So when I put them in the quadratic formula, do I disregard one answer or take both as when its going up and then going down?
    Good question. I thought about that when I answered your post. I would tend to say that the question is concerned about the upward motion, but I wouldn't neglect the downward.

    I would use both, but I would explain my answer to avoid confusion and cover my bases.
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