Earth has GM=3.986×10¹⁴ m³/s² and radius 6371 km. Find the minimum instantaneous velocity increment, in km/s, for a spacecraft in a circular orbit at altitude 4000 km to escape Earth (two decimal places).
The orbital radius is r=(6371+4000)×1000 m. Circular and escape speeds there are $\sqrt{GM/r}$ and $\sqrt{2GM/r}$. A prograde tangential impulse minimizes the required increment: $\Delta v=(\sqrt2-1)\sqrt{GM/r}=2.568$ km/s, or 2.57.
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