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Falling object (gravity + kinematics)

  1. Dec 20, 2006 #1
    1. The problem statement, all variables and given/known data

    An object is dropped from an altitude of one Earth radius above Earth's surface. If M is the mass of Earth and R is its radus, find the speed of the object just before it hits Earth.

    2. Relevant equations

    [tex]F_g=ma_g=\frac{GMm}{r^2}[/tex]

    [tex]v^2=v_0^2+2ah[/tex]

    3. The attempt at a solution

    [tex]F_g=\frac{GMm}{(2R)^2}=ma_g[/tex]

    [tex]a_g=\frac{GM}{4R^2}[/tex]

    Now, plug that into the kintematics equation and get

    [tex]v^2=0+2(\frac{GM}{4R^2})R[/tex]

    [tex]v=\sqrt{\frac{GM}{2R}}[/tex]

    But the correct answer is given as [tex]\sqrt{\frac{GM}{R}}[/tex], and I can't find my error.
     
  2. jcsd
  3. Dec 20, 2006 #2

    Doc Al

    User Avatar

    Staff: Mentor

    Note that F_g and a_g are not constant, but are functions of r.
    But this kinematic equation assumes constant acceleration.

    That's only the acceleration at the point r = 2R; as the object falls, the acceleration increases.

    Your error is treating this as a constant acceleration problem. Instead of using kinematics, why not use energy conservation? (What's the general form for gravitational PE? Note that "mgh" is only valid near the earth's surface--no good here.)
     
  4. Dec 20, 2006 #3
    Thanks, got it now!
     
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