AP Physics 1

Ratio of Orbital Radii

Satellites 1 and 2 both orbit a planet in circular orbits at different radii. Satellite 1 has twice
the speed of Satellite 2. Determine the ratio R1/R2 of the orbital radii.



Typed Out Solution


What is Given

Given by
the Problem
What we
Know


$V_1 = 2V_2$



$F_c = \frac{M_sV^2}{R}$
$F_g = \frac{GM_pM_s}{R^2}$



Calculations


Find a General Relationship
for the orbital speed of an object

$F_c = F_c$
$ \frac{M_sV^2}{R} = \frac{GM_pM_s}{R^2}$
Note that $M_s$ cancels out
as does one of the R's
$V^2 =\frac{GM_p}{R}$
Which is a general Relationship
for the Orbital Velocity

 

Satellite 1 Satellite 2

$V_1^2 = \frac{GM_p}{R_1}$
Solve for $R_1$

$V_2^2 = \frac{GM_p}{R_2}$
Solve for $R_2$

$R_1 = \frac{GM_p}{V_1^2}$
Substitute $V_1 = 2V_2$ for $V_1$

$R_2 = \frac{GM_p}{V_2^2}$


$R_1 = \frac{GM_p}{(2V_2)^2}$



$\frac{R_1}{R_2} = \frac{ \frac{GM_p}{4V_2^2}}{\frac{GM_p}{V_2^2}}$


Notice that most everything cancels out, except for a 1 and the 4

$R_1 = \frac{GM_p}{4V_2^2}$

 


$\frac{R_1}{R_2} = \frac{1}{4}$
or $R_2 = 4R_1$