Gravity Equation:
[tex]f = \frac{gm _{0}(m) }{ {r}^{2} } [/tex]
Gravitational Force relative to Moon
[tex]f = \frac{g_{moon} m _{object}(m_{moon} ) }{ {r}^{2} } [/tex]
Gravitational Force relative to earth:
[tex]f = \frac{g_{earth} m _{object}(m_{earth} ) }{ {r}^{2} } [/tex]
Equated Forces such that the gravitational forces are equal:
[tex] \frac{g_{earth}m _{object}(m_{earth} ) }{ {(r)}^{2} } = \frac{g_{moon}m _{object}(m_{moon} ) }{ {(d - r)}^{2} } [/tex]
cancel m_object and g as it is universal.
[tex] \frac{ (m_{earth} ) }{ {(r)}^{2} } = \frac{(m_{moon} ) }{ {(d - r)}^{2} } [/tex]
Some values for our variables:
[tex]d= 384.4 \times {10}^{6} m \\ m _{moon} = 7.35 \times {10}^{22}kg \\ m_{earth} = 5.98 \times {10}^{24} kg[/tex]
pretty it up:
[tex] \frac{ (m_{earth} ) }{ {(r)}^{2} } = \frac{(m_{moon} ) }{ {(d - r)}^{2} } \\ \\ \frac{ {(d - r)}^{2} }{ {(r)}^{2} } = \frac{(m_{moon} ) }{ m_{earth} } \\ \\ \sqrt{ \frac{ {(d - r)}^{2} }{ {(r)}^{2} }} = \sqrt{ \frac{(m_{moon} ) }{ m_{earth} } } \\ \\ \frac{ {d - r} }{ {(r)}} = \sqrt{ \frac{(m_{moon} ) }{ m_{earth} } } \\ [/tex]
Now plug in:
[tex] \frac{ {(d - r)} }{ {(r)}} = \sqrt{ \frac{(m_{moon} ) }{ m_{earth} } } \\ \\ \frac{ { 384.4 \times {10}^{6} - r} }{ {r }} = \sqrt{ \frac{(7.35 \times {10}^{22} )} { (5.98 \times {10}^{24} )}} \\ \\ \frac{ {384.4 \times {10}^{6} - r} }{ {r}} = 9[/tex]
solve for r:
[tex] \frac{(384.4 \times {10}^{6} )}{r} - 1 = 9 \\ \\ r = 3.844 \times {10}^{7} [/tex]
so the distance is about 3.844x10^7 meters or about 24,000 miles
