Let's consider a spacecraft's motion orbiting the Moon in two$-$dimensional space $($Fig$-$

ure $1).$ The spacecraft is in a circular orbit around the Moon at a distance of $r_M$ and

a constant orbital angular speed of ${}_M.$ The Moon is in a circular orbit around the

Earth at a distance of $r_E$ and a constant orbital angular speed of ${}_E.$ Compute the

position, velocity, and acceleration of the spacecraft with respect to the Earth. There

are six unit vectors to be used. vec$(e{)}_{Er}$ and vec$(e{)}_E$ are unit vectors toward the Moon from

the Earth and the tangential direction, respectively. vec$(e{)}_{Mr}$ and vec$(e{)}_M$ are those toward the

spacecraft from the Moon and the tangential direction, respectively. Finally, vec$(e{)}_x$ and

vec$(e{)}_y$ define the inertial frame. Hint: Use the transport theorem, time derivatives with

respect to a proper frame, and dot products of unit vectors for angles between them.

Consider all the motions to be on a flat plane.

$($a$)(5$ pt$)$ Position with respect to the Earth in the inertial frame.

$($b$)(5$ pt$)$ Velocity with respect to the Earth in the inertial frame. Hint: The velocity

means a time derivative of the position with respect to the inertial frame.

$($c$)(5$ pt$)$ Acceleration with respect to the Earth in the inertial frame. Hint: Use a

time derivative with respect to the inertial frame.