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Falling object (gravity + kinematics)

  • Thread starter mbrmbrg
  • Start date
  • #1
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Homework Statement



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.

Homework Equations



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

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

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.
 

Answers and Replies

  • #2
Doc Al
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[tex]F_g=ma_g=\frac{GMm}{r^2}[/tex]
Note that F_g and a_g are not constant, but are functions of r.
[tex]v^2=v_0^2+2ah[/tex]
But this kinematic equation assumes constant acceleration.

The Attempt at a Solution



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

[tex]a_g=\frac{GM}{4R^2}[/tex]
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.)
 
  • #3
486
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Thanks, got it now!
 

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