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Conservation of energy as a function of Time

  1. Sep 7, 2007 #1
    1. The problem statement, all variables and given/known data

    A ball of unspecified mass is in a free fall - and we are supposed to rearrange equations to teach us the basics of kinematics. It's a general question with no values given for the variables

    Use your conservation of energy equation: v = sqrt(2gh) and calculus to write an expression for the ball's position as a function of time.

    2. Relevant equations

    It's the last part that is getting me. I'm thinking they want me to use an integral to find the position-time graph or something. Alternatively, I can't figure out how to relate a kinematic equation: y=(1/2)gt^2 (in this case) to the conservation of energy equation we derived.

    (We derived this conservation of energy equation from putting the kinetic energy in terms of its velocity - then relating this energy transfer to the force of gravity working on this free falling ball. The result was the above equation - which is supposed to be an equation of velocity in terms of the forces acting on the ball. Did I do that right?


    3. Attempt at Solution

    I've been bothered by this for hours. I tried writing the integral of the v = sqrt(2gh) function, but I just get a messy integral and I'm not sure that's the right idea anyway. Maybe I'm misinterpreting the question?

    Thanks for any help you can give!!!
     
  2. jcsd
  3. Sep 7, 2007 #2
    Okay let's write v = sqrt(2gh), in calculus terms

    [tex]\frac{dy}{dt} = \sqrt{2gy(t)[/tex]

    Or

    [tex]\frac{dy}{\sqrt{2gy(t)}} = dt[/tex]

    Or if you integrate both sides

    [tex]\int{\frac{dy}{\sqrt{2gy(t)}}} = \int{dt} = t + C[/tex]


    This is an easy enough integral, then solve for y(t) and use your initial conditions to find C.
     
  4. Sep 7, 2007 #3

    andrevdh

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    Homework Helper

    Try using

    [tex]v = \frac{dy}{dt}[/tex]

    and

    [tex]y(t)=h_o - h(t)[/tex]

    y positive downwards.
     
  5. Sep 7, 2007 #4
    Excellent! Thank you - that makes great sense!
     
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