Question

A spaceship travelling between the earth and the moon.the distance between the center of the earth and the moon is 3.8x10⁸m.calculate the distance of spaceship from the earth when it experiences zero gravitational force.(mass of the earth=5.97x10²⁴kg;mass of the moon =7.35x10²²kg)

Answer 1

In order to spaceship experiences zero gravitational force, the gravitational forces exerted on the spaceship by the Earth and Moon must be equal:

$$F_E=F_M,$$ $$\dfrac{GM_EM_{ship}}{r^2}=\dfrac{GM_MM_{ship}}{(R_{EM}-r)^2},$$ $$\dfrac{M_E}{r^2}=\dfrac{M_M}{(R_{EM}-r)^2}.$$

From this equation we can find the distance of spaceship from the Earth when it experiences zero gravitational force:

$$r=\dfrac{R_{EM}\sqrt{\dfrac{M_E}{M_M}}}{(1+\sqrt{\dfrac{M_E}{M_M}})},$$ $$r=\dfrac{3.8\cdot10^8\ m\cdot\sqrt{\dfrac{5.97\cdot10^{24}\ kg}{7.35\cdot10^{22}\ kg}}}{(1+\sqrt{\dfrac{5.97\cdot10^{24}\ kg}{7.35\cdot10^{22}\ kg}})}=3.4\cdot10^8\ m.$$