At what speed should a space probe be fired from the earth if it is required to still be traveling at a speed of 3.48 km/s, even after coasting to an exceedingly great distance from the planet (a distance that is essentially infinite)?

Answer 1

Figure the work it takes to move an object from Earths surface to infintity, then add the remaining KE you want.

Work=INT GMe*m dr/r^2 + m 3.48^2

Set that equal to the KE at Earths Surface, mv^2, and solve for v.