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Calculus

Suppose you throw an object from a great height, so that it reaches very nearly terminal velocity by time it hits the ground. By measuring the impact, you determine that this terminal velocity is -49 mi sec.
A. Write the equation representing the velocity v(t) of the object at time t seconds given the initial velocity v0 and the fact that acceleration due to gravity 9.8 m/sec2. (Here, assume you're modeling the falling body with the differential equation dy/dt = g-kv, and use the resulting formula or v(t) found in the Tutorial. Of course, you can derive it if you'd like.)
B. Determine the value of k, the "continuous percentage growth rate" from the velocity equation, by utilizing the information given concerning the terminal velocity.
C. Using the value of k you derived above, at what velocity must the object be thrown upward if you want it to reach its peak height after 3 sec? Approximate your solution to three decimal places, and justify your answer.

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  1. v = dy/dt
    so, if v = g-kv
    this is just a linear DEm with solution
    v = ce^(-3t) + g/k
    since v(0) = v0,
    c = v0 - g/k

    That may not look exactly like the formula you derived, but it is the same, up to assigning the constants.

    for large t, e^(-3t) = 0, so we have
    g/k = -49

    Now you can answer C.

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    oobleck
  2. hey oobleck... what?

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