physics
posted by Jake .
i posted this earlier but am now stuck on getting through the algebra
1. The problem statement, all variables and given/known data
13) (II) At what distance from the Earth will a spacecraft on the way to the Moon experiance zero net force due to these two bodies becasue the Earth and Moon pull with equal and opposite forces?
2. Relevant equations
NET F = ma
G = 6.67 E11 (Nm^2)/kg^2
Fg = (GmM)/r^2
Mass Moon = 7.35 E 22 kg
Mass Earth = 5.98 E 24 Kg
r Earth to Moon = 384,403,000 m
3. The attempt at a solution
Apply Newton's s second law in the radial direction
NET F = m_craft( a_radial) = Fg moon = Fg Earth = 0
= (G m_craft m_moon)/(384,403,000 m  r)^2 = (G m_craft m_Earth)/r^2
m_craft cancels
G cancels
m_moon/(384,403,000 m  r)^2 = m_Earth/r^2
simplify
m_moon/((384,403,000 m)^2 r^2) = m_Earth/r^2
raise both sides to negative one power
((384,403,000 m)^2 r^2)/m_moon = r^2/m_Earth
multiply both sides by m_Earth
m_Earth( (384,403,000 m)^2  r^2 )/m_moon = r^2
simplify
( m_Earth(384,403,000 m)^2  m_Earth(r^2) )/m_moon = r^2
simplify further
(m_Earth * (384,403,000 m)^2)/m_moon  (m_Earth (r^2) )/m_moon = r^2
add (m_Earth (r^2) )/m_moon to both sides
(m_Earth * (384,403,000 m)^2)/m_moon = (m_Earth (r^2) )/m_moon + r^2
I don't know were to go from here...

I believed this is correct after noticing my mistake
Ok I'm here what do I do now
0 = (m_M r^2)/m_E + r^2 + 2(3.84403 m)r  (3.84403E9 m)2 
sorry
0 = (m_M r^2)/m_E + r^2 + 2(3.84403 m)r  (3.84403E9 m)^2 
sorry again
0 = (m_M r^2)/m_E + r^2 + 2(3.84403 E 8 m)r  (3.84403 E8 m)^2
sorry about that 