What Is the Minimum Distance Between a Slow-Launched Satellite and Its Planet?

[slai13]

Homework Statement

Two satellites are launched at a distance R from a planet of negligible radius. Both satellites are launched in the tangential direction. The first satellite launches correctly at a speed v0 and enters a circular orbit. The second satellite, however, is launched at a speed .5v0. what is the minimum distance between the second satellite and the planet over the course of its orbit?

R=launch radius, r=minimum radius, v=velocity at minimum radius

Homework Equations

F=GMm/R^2
U= -GMm/R
K=.5mv^2
mvr= const. (conservation of angular momentum)
K+U=const. (conservation of energy)

The Attempt at a Solution

GM= R(v0)^2
v0R/2 = vr, v = (v0R)/(2r)
.5m(.5v0)^2 - GM (m/R) = .5m(v^2) - GM (m/r)

Substituting in values and solving for r doesn't lead me to the answer. Any help?

 

[TSny]

Hi, slai13. Welcome to PF!

Everything looks good to me. Assuming your algebra is correct, I think you should get the right answer. Can you tell us what you got?

 

[slai13]

Sorry to have wasted your time.

I worked out the quadratic equation and got R/7 and R. R/7, I'm sure, is the minimum I'm looking for. Thanks!

 

[TSny]

No problem. Good work!