Question

A spaceship is on a straight-line path between Earth and the Moon. At whatdistance from Earth is the net gravitational force on the spaceship zero?

Short answer

Net gravitational force on the spaceship is zero at a distance of$3.44\times {10}^8\mathrm{m}$ from the earth.

Step-by-step solution

Listing the given quantities

The distance between the Earth and the Moon is ${\mathrm{R}}_{\mathrm{Earth}}=3.82\times {10}^8\mathrm{m}$.

Understanding the concept of gravitational force

Let the distance from Earth to the spaceship be r.

Formula:

$\begin{array}{rcl}{\mathrm{f}}_{\mathrm{E}}& =& \frac{{\mathrm{GM}}_{\mathrm{e}}\mathrm{m}}{{\left({\mathrm{R}}_{\mathrm{em}}-\mathrm{r}\right)}^2}\\ & =& \frac{{\mathrm{GM}}_{\mathrm{e}}\mathrm{m}}{{\mathrm{r}}^2}\end{array}$

where m is the mass of the spaceship.

Explanation

Solving for r, we obtain

$\begin{array}{rcl}\mathrm{r}& =& \frac{{\mathrm{R}}_{\mathrm{em}}}{\sqrt{\mathrm{M}/{\mathrm{M}}_{\mathrm{e}}}+1}\\ & =& \frac{3.82\times {10}^8\mathrm{m}}{\sqrt{\left(7.36\times {10}^{22}\mathrm{kg}/\left(5.98\times {10}^{24}\right)\mathrm{kg}\right)}+1}\\ & =& 3.44\times {10}^8\mathrm{m}\end{array}$

The net gravitational force on the spaceship is zero at a distance of$3.44\times {10}^8\mathrm{m}$ from the earth.