For a spacecraft going directly from the earth to the moon, beyond what point will lunar gravity begin to dominate? that is, where will the lunar gravitational force be equal in magnitude to the earth’s gravitational force? are the astronauts onboard a spacecraft at this point truly weightless?

3.46 x 10^8 m

Explanation:

Let the distance from the earth be r, total distance from earth to moon be d,

then distance from the moon to the point of equal gravity= d -r.

Mass of the earth, M_E = 598 x 10^22 kg

Mass of the moon, M_m= 7.36 x 10^22 kg

Distance of the mood from the earth, d = 3.85 x 10^8 m

Force of gravity exerted on a mass, m by the earth,

             F_E = G. M_E×m/r^2

Force of gravity exerted on a mass, m by the moon,

              F_m = G×M_m×m/(d-r)^2

These are equal,  So

 G×M_E.m/r^2 =G×M_m.m/(d-r)^2

     [r/(d - r)]^2=  M_E/M_m

    = 598/7.36 = 81.25

      r/(d - r) = √81.25 = 9.014

      9.014×(d - r) = r

    10.014.r = 9.014 x 3.85 x 108

Hence,    r= 9.014 x 3.85 x 108 /10.014

             = 3.46 x 10^8 m