What is the minimum initial velocity necessary for an object to leave Earth?
The problem can be easily solved using energy formulas. The only force that acts on the departing object is the gravitational force, which is conservative. Therefore the total energy is conserved on the trajectory:E=mv2/2-GmM/r=ct.The energy on the surface of the planet is:E=mv2i/2-GmM/R where vi is the initial velocity and R is the radius of Earth.At infinity(where the objects eventually gets since it leaves Earth):E=mv2f/2 where vf is the final velocity, which will be set to 0 in order to minimise the initial velocity.Equating the energies of the two positions we get:mv2i/2-GmM/R=0vi=(2GM/R)1/2 After introducing the values for the gravitational constant, mass and radius of Earth we get the final velocity:vi=11.2 km/s
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