Energy of a SpacecraftVery far from earth(at R=\infty), a spacecraft has run out of fuel and its kineticenergy is zero. If only the gravitational force of the earth wereto act on it (i.e., neglect the forces from the sun and other solarsystem objects), the spacecraft would eventually crash into theearth. The mass of the earth is M_e and its radius is R_e. Neglect air resistance throughout this problem, sincethe spacecraft is primarily moving through the near vacuum ofspace.Part AFind the speed s_e of the spacecraft when it crashes into the earth.Express the speed in terms ofM_e, R_e, and the universal gravitational constant G.s_e =

In this case mechanical energy is conserved, which means that the sum of the initial kinetic energy and initial potential gravitational energy will be equal to the sum of the final kinetic energy and final potential gravitational energy:

[tex]K_i+U_i=K_f+U_f[/tex]

Which in our case will be:

[tex]\frac{mv_i^2}{2}+\frac{-GM_em}{r_i^2}=\frac{mv_f^2}{2}+\frac{-GM_em}{r_f^2}[/tex]

Which, since [tex]v_i=0m/s[/tex], [tex]r_i=infinity[/tex], [tex]r_f=R_e[/tex], [tex]v_f=s_e[/tex] and canceling m means that:

[tex]\frac{s_f^2}{2}=\frac{GM_e}{R_e^2}[/tex]

Solving for the final velocity we get:

[tex]s_e=\sqrt{\frac{2GM_e}{R_e^2}}[/tex]