What is d min energy required 2 launch a satellite of mass m from d surface of a planet of mass M & radius R in a circular orbit at an altitude of 2R?
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PE of satellite at altitude 2 R = - G M m / 3 R
KE of satellite in orbit of radius 3R = 1/2 m v²
m v² / R = centripetal force = G m M / R²
v² = G M / R
Total Energy of satellite in orbit = - GM m / 3R + 1/2 m G M /R
= - 2 G M m / 3 R
This is the minimum energy required to launch the satellite
KE of satellite in orbit of radius 3R = 1/2 m v²
m v² / R = centripetal force = G m M / R²
v² = G M / R
Total Energy of satellite in orbit = - GM m / 3R + 1/2 m G M /R
= - 2 G M m / 3 R
This is the minimum energy required to launch the satellite
total energy on surface of Planet M: KE - GMm/R. PE of m in orbit: -GMm/3R. KE in orbit +GMm/6R. Total Energy in orbit = 1/2 PE. Energy required: GMm/R(1-1/6) = 5/6 GMm/R
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