Space probe between the earth and moon

[toothpaste666]

Homework Statement

The mass of the Earth is Me and the mass of the Moon is Mm. The (center to center) Earth to moon distance is d.
a) If a space probe is sent directly from the Earth to the moon, how far from the center of the Earth would the net gravitational force (due to the Earth and moon) on the probe be zero?
b) Neglecting friction, how much net energy would it take to lift a space probe from the surface of the Earth to the moons surface?

Homework Equations

[itex] E = K + U [/itex]

[itex] F = G\frac{Mm}{r^2} [/itex]

The Attempt at a Solution

a) There are two forces acting on the probe, the force of gravity of the moon and the force of gravity of the earth. We are trying to find the distance where the combination of these two forces is 0 so:

[itex] M_e [/itex] = mass of Earth [itex] M_m [/itex] = mass of moon [itex] M_p [/itex] = mass of probe [itex] d [/itex] = Earth to moon distance [itex] r [/itex] = Earth to probe distance

[itex] F_{moon} - F_{earth} = 0 [/itex]

[itex] F_{moon} = F_{earth} [/itex]

[itex] G\frac{M_m M_p}{(d-r)^2} = G\frac{M_e M_p}{r^2} [/itex]

[itex] \frac{M_m}{(d-r)^2} = \frac{M_e}{r^2} [/itex]

It seems I have hit a wall. How can I isolate r? b)

since we need the satellite to escape Earth's orbit we need to find the escape velocity. this happens when the satellite goes to infinity with a speed of 0.

[itex] \frac{v_{esc}^2}{2} - G\frac{M_e}{r_e} = 0 + 0 [/itex] [itex] \frac{v_{esc}^2}{2} = G\frac{M_e}{r_e} [/itex]

[itex] v_{esc} = (2G\frac{M_e}{r_e})^\frac{1}{2} [/itex]

now we need to find the energy needed

[itex] W + \frac{v_{esc}^2}{2} - G\frac{M_e}{r_e} = 0 [/itex]

[itex] W = - \frac{v_{esc}^2}{2} + G\frac{M_e}{r_e} [/itex]

would this give the correct answer?

 

[paisiello2]

a) Try inverting the equation and then expand the d-r term; then find the roots of the equation.

b) The question didn't ask for the escape velocity, so I think you should drop this term; however, you need to account for the fact that the moon will reduce the work required because it attracts the satellite.

 

[haruspex]

Your method in part b) appears to guarantee the answer W = 0.
Since it asks for net energy, you need to take into account the potential energy when on the surface of the moon.