Question

(a) Imagine a straight line connecting the center of the moon with the center of the earth. An object moving along this line would feel two main gravitational forces: A force pulling it towards the center of the earth, and a force pulling it towards the center of the moon. Find the distance along this line, measured from the center of the Moon, such that the net force is zero.

(b). The moon-dwellers launch a rock from the surface of the moon with just as much speed as is necessary to reach this point. Find this speed. (Use conservation of energy, taking into account gravitational potential energy due to both the moon and the earth.)

(c). Suppose that a rock is launched from the surface of the moon with a speed slightly greater than that given in (2). What will be the speed of the rock when it reaches the earth?

Answer 1

a)Imagine a straight line connecting the center of the moon

so at that point

{Fmoon = Fearth}

It's a equal Fmoon and Fearth

G mmoon/x^2 = G mearth/ (R -x)^2

where R is the distance from eath to moon

7.35E22/x^2

= 5.97E24/(3.84E8-x)^2

Also find x

x=3.84E7 m

b)The moon-dwellers launch a rock from the surface of the moon

then 1/2 mv^2 - G m moon/r - G mearth m/( R-r) = - G m mmoon/x - G mearth m/( R-x)

0.5*v^2 - 6.67E-11*7.35E22/1.737E6 - 6.67E-11*5.97E24/(3.84E8-1.737E6)=- 6.67E-11*7.35E22/3.84E7 - 6.67E-11*5.97E24/(3.84E8-3.84E7)

Also find on

v=2273 m/s

c)Suppose that a rock is launched from the surface of the moon

if it makes it all the way to earth then

1/2 mv^2 - G m mearth/rearth - G m mmoon/( R - rearth)

= - G m mmoon/x - G mearth m/( R-x)

0.5*v^2 - 6.67E-11*7.35E22/(3.84E8-6.378E6) - 6.67E-11*5.97E24/(6.378E6)

=- 6.67E-11*7.35E22/3.84E7 - 6.67E-11*5.97E24/(3.84E8-3.84E7)

Also find on v

v=11060 m/s