A spacecraft (mass = m) orbits a planet (mass = M) in a circular orbit (radius = R). What is the minimum energy required to send this spacecraft to a distant point in space where the gravitational force on the spacecraft by the planet is negligible?
A) GmM/(4R)
B) GmM/R
C) GmM/(2R)
D) GmM/(3R)
E) 2GmM/(5R)

Question

Quizplus · Accepted Answer

The minimum energy required to send the spacecraft to a distant point in space where the gravitational force on the spacecraft by the planet is negligible is equal to the difference in potential energy between the initial circular orbit and the final distance where the force is negligible. At a distance where the gravitational force is negligible, the potential energy of the spacecraft is zero. Therefore, the minimum energy required is equal to the initial potential energy, which is given by U = -GmM/R. To convert this potential energy to kinetic energy, we can use the conservation of energy principle:

(1/2)mv^2 = GmM/R
v^2 = 2GM/R
v = sqrt(2GM/R)

Therefore, the minimum energy required is equal to the kinetic energy of the spacecraft in its initial circular orbit:

K = (1/2)mv^2 = (1/2)m(2GM/R) = GmM/R

Therefore, the correct answer is C, GmM/(2R).

A Spacecraft (Mass = M) Orbits a Planet (Mass =