Question

Mars has a mass of about 6.58 × 1023 kg, and its moon Phobos has a mass of about 9.3 × 1015 kg. If the magnitude of the gravitational force between the two bodies is 4.18 × 1015 N, how far apart are Mars and Phobos? The value of the universal gravitational constant is 6.673 × 10−11 N · m2 /kg2 . Answer in units of m.

Answer 1

This looks complicated, but it's actually not too tough.

The formula for the gravitational force between two objects is

              Force = G  (one mass) (other mass) / (distance²) .

The question GAVE us all of those numbers except the distance.
All we have to do is pluggum in, massage it around, and find
the distance. 

Force  =         4.18 x 10¹⁵     N
  G  =             6.673 x 10⁻¹¹  N·m²/kg²
One mass =   6.58 x 10²³     kg
Other mass = 9.3 x 10¹⁵       kg   .

The only tricky thing about this is gonna be the arithmetic ...
keeping all the exponents straight.

Take the formula for the gravitational force and plug in
everything we know:

Force = (G) · (one mass) · (other mass) / (distance²) 

4.18x10¹⁵N = (6.673x10⁻¹¹N-m²/kg²)·(6.58x10²³kg)·(9.3x10¹⁵kg) / (distance²).

Multiply each side by  (distance²):

(distance²)·(4.18x10¹⁵N) = (6.673x10⁻¹¹N-m²/kg²)·(6.58x10²³kg)·(9.3x10¹⁵kg) 

Divide each side by  (4.18 x 10¹⁵ N) :

(distance²)=(6.673x10⁻¹¹N-m²/kg²)·(6.58x10²³kg)·(9.3x10¹⁵kg) / (4.18x10¹⁵N)

That's the end of the Physics and Algebra.  The only thing left is Arithmetic.
We have to simplify that whole ugly thing on the right side of the equation,
and then take the square root of each side.

When I crunch down the right side of that equation, I get

           (distance²)  =  9.769 x 10¹³  m²

and when I take the square root of each side, I get

             distance  =  9.884 x 10⁶ meters .   **

You should check my Arithmetic.   **
(Pause occasionally to let your calculator cool off.)


BY THE WAY ... 
That "distance" in the equation for gravitational force is the distance
between the CENTERS of the two objects. 
This doesn't make much difference for Phobos, because Phobos isn't
much bigger than a big sweet potato.  But it does make a difference for
Mars. 
The 'distance' we find with all of this nonsense is NOT the distance
between Phobos and the surface of Mars.  It's the distance between 
Phobos and the CENTER of Mars, so it includes the planet's radius.   


** Consulting online resources between Floogle and Flickerpedia,
I found that the orbital distance of Phobos from Mars varies between
9,234 km and 9,517 km.  Add the planet's radius to these, and I'm
beginning to feel confidence in the results of my back-of-the-napkin
calculation.  But you should still check my Arithmetic.