need help with equilibrium question!

**jddery** · November 27th 2009, 06:45 AM

A satellite probe is put into an orbit around a distant planet such that at apoapsis its orbital radius is 4989 km and at periapsis its orbital radius is 2959 km. If it has a speed of 1.36 km/s at apoapsis what is its speed at periapsis?

[Specify units of 'km/s' or 'm/s' in your answer (without quotes).]

**mathaddict** · November 27th 2009, 07:52 AM

Originally Posted by jddery

A satellite probe is put into an orbit around a distant planet such that at apoapsis its orbital radius is 4989 km and at periapsis its orbital radius is 2959 km. If it has a speed of 1.36 km/s at apoapsis what is its speed at periapsis?

[Specify units of 'km/s' or 'm/s' in your answer (without quotes).]

HI

In an orbital system , the gravitational force=centripetal force from the energy principles again .

$\frac{GMm}{r^2}=\frac{mv^2}{r}$ ,

where G is the gravitational constant , M is the mass of planet , m : mass of satelite , r : orbitral radius

$v^2=\frac{GM}{r}$

Since GM are both constants , $v^2\propto \frac{1}{r}$

$v\propto \frac{1}{\sqrt{r}}$

At apoapsis , r=4989 , v=1.36

At periapsis , r=2959 , v=??

Now substitute appropriately ,

$1.36 \propto \frac{1}{\sqrt{4989}}$ ----- 1

$v\propto \frac{1}{\sqrt{2959}}$ ---- 2

Divide them to get v which is approximately 1.77 km/s

**skeeter** · November 27th 2009, 08:36 AM

Originally Posted by jddery

A satellite probe is put into an orbit around a distant planet such that at apoapsis its orbital radius is 4989 km and at periapsis its orbital radius is 2959 km. If it has a speed of 1.36 km/s at apoapsis what is its speed at periapsis?

[Specify units of 'km/s' or 'm/s' in your answer (without quotes).]

ar periapsis and apoapsis, the velocity and position vectors are perpendicular. use conservation of angular momentum ...

$mv_ar_a = mv_br_b$

$v_a = \frac{v_b r_b}{r_a}$

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