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How to solve a velocity problem as an initial condition problem

  1. Sep 8, 2013 #1
    1. The problem statement, all variables and given/known data

    A small rock is thrown vertically upward with a speed of 17.0m/s from the edge of the roof of a 30.0m tall building. The rock doesn't hit the building on its way back down and lands in the street below. Air resistance can be neglected.

    2. Relevant equations

    Acceleration of gravity = 9.81m/s^2

    The normal way:
    v^2 = u^2 + 2as
    v=u+a*t
    http://answers.yahoo.com/question/index?qid=20120204155622AAIs1sA

    How I want to do it:
    dv/dt = a(t)
    ds/dt = v(t)

    3. The attempt at a solution
    a(t) = -9.8 m/s^2
    v(t) = -9.8t m/s + C(m/s)
    17 = -9.8*0 + C, C=17
    v(t) = -9.8t(m/s) + 17(m/s)
    s(t) = -4.9t^2 + 17t + C
    30 = -4.9*0 + 17*0 + C
    C = 30
    s(t) = -4.9t^2+17t+30
    -30 = -4.9t^2+17t+30

    t=5.64s for part B.
    For part A, this would make it -9.8*5.64 + 17 = -38.272.


    My answers are wrong. Using the equations listed at Yahoo answers:
    v=(17^2+2*9.81*30)^(1/2)=29.6m/s and 29.62=-17+9.81*t, t=4.76s

    I'm certain there's a way to solve this as initial value problems. I'd rather get my calculus on than try to memorize these equations!
     
  2. jcsd
  3. Sep 8, 2013 #2
    Why -30 on the left hand side?
     
  4. Sep 8, 2013 #3
    I believe a bad guess. The initial position is y=0m (the level of the roof), the ball ends up at y=-30.

    That probably should be a zero.
     
  5. Sep 8, 2013 #4
    You need to decide. Either s(0) = 30, which fixes one value for the integration constant. Or s(0) = 0, which fixes some other value. Then you should have s(T) = 0 or S(T) = -30, respectively. But s(0) = 30, which you chose originally, is not compatible with with s(T) = -30 (because that would mean the street is 60 meters below the 30 meter tall building, which is impossible).
     
  6. Sep 8, 2013 #5
    Thank you! That does provide the correct answer in either case (t=4.76, then that can be substituted into v(t) = -29.6m/s!)

    I'm glad to know it was something simple.
     
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