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One of the first exercises in Arnold's mechanics book is

Problem. Determine with what velocity a stone must be thrown in order that it fly infinitely far from the surface of the earth. Answer. $\geq 11.2$ km/sec.

To this point the only thing presented about the system is a governing differential equation:

$$\ddot x=-g\frac{r_0^2}{(r_0+x)^2}$$

I don't know how to solve this differential equation. I anticipate there is a way to solve the problem without solving the DE directly, instead somehow producing an expression which gives x's velocity "at time infinity" and then solving for the critical initial condition. How can this be done?

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$ \ddot x= \frac{dv}{dt}= \frac{dx}{dt} \cdot\frac{dv}{dx}=v\frac{dv}{dx} $

So integrating from the surface of the Earth to infinity ...

$$ \int _{v_o}^{v_f} v dv =-gr_0^2 \int _0 ^\infty \frac{dx}{ (r_0 + x)^2} $$

To get the slowest velocity solve for $v_0$ when $v_f=0$. You should get $v_o = \sqrt{2gr_0} $

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