At what speed should a space probe be fired from the Earth if it is required to still be travelling at a speed?

At what speed should a space probe be fired from the Earth if it is required to still be travelling at a speed of 6.75 km/s, even after coasting to an exceedingly great distance from the planet (a distance that is essentially infinite)? (Neglect the friction and drag in the atmosphere.)








Please also look at my other questions. Any help is appreciated.|||we can approach this via energy conservation





at the surface of the earth, the object has KE = 1/2 m v0^2 and PE = - GMm/R where G is the newtonian grav cst, M the mass of the earth, m the mass of the spacecraft and R the radius of the earth





at great distance from the earth, the object has KE = 1/2 m V^2 where V = 6750m/s and PE = 0 since it is infinitely far from the earth





we have


1/2 m V^2 = 1/2 m v0^2 - GMm/R





multiply through by 1/2 m and get





V^2 = v0^2 -2GM/R





v= Sqrt[V^2 +2GM/R] = Sqrt[6750^2 + 6.67x10^-11x5.98x10^24kg/6.4x10^6]


v = 13046m/s|||..or since Ek var as v^2..then energy @ take off must've been prop 2 (escapeVel^2+probeVel^2)


Ve=11.183km/sec, probevel=6.75km/sec..so tot of sqres=170.62..take roots %26gt;V start=13km/sec..roughly