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**1. Problem:**A particle falls to Earth starting from rest at a great height (may times Earth's radius ). Neglect air resistance and show that the particle requires approximately 9/11 of the total time of fall to traverse the first half of the distance.

## Homework Equations

[tex]F=\frac{-G M_e m}{r^2}[/tex]

## The Attempt at a Solution

[tex]F=\frac{-G M_e m}{r^2}[/tex]

[tex]a=\frac{-G M_e}{r^2}[/tex]

[tex]\frac{\partial v}{\partial r} v = \frac{-G M_e}{r^2}[/tex]

seperate and integrate

[tex]

\frac{1}{2}v^2=\frac{G M_e}{r}+K[/tex]

[tex]v^2 = \frac{2G M_e}{r}+K[/tex]

[tex]v^2(r_0)=\frac{2G M_e}{r_0}+K=0[/tex]

[tex]v^2=2G M_e \( \frac{1}{r}-\frac{1}{r_0}\)[/tex]

Then take square root, replace v with the time derivative of r, seperate and integrate and you get this mess:

[tex]\int{\sqrt{\frac{r_0 r}{r_0-r}}\partial r}=\sqrt{G M_e}t[/tex]

Which comes out to having a tangent in it, and in general very messy. Also when you plug the two values [tex]r_0[/tex] and 0, you wind up with an infinity. So any ideas where I went wrong?

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