Escape velocity

As we have seen, an object that is hanging in space near the Earth will have negative gravitational energy. A satellite has kinetic energy, making the total energy less negative, but it is still negative.

When we say we want an object to “escape” the Earth’s gravitational pull, we mean that it will go to $r=\infty$ and never come back. To accomplish this, we must have $E_{\text{tot}}=0\text{.}$ This occurs when

$E_{\text{k}}>G\frac{m_1{m}_2}{r}\text{,}$

and if we solve for speed, we find that the escape velocity is given by

$v_{\text{esc}}=\sqrt{\frac{2G{m}_{\text{planet}}}{r}}\text{.}$