Gravitation: Satellite Orbits

[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!

 

[Nickie]

I would suggest approaching this problem by first considering the basic principles of gravitation and orbital motion. The first satellite is launched at a tangential speed that allows it to enter a circular orbit, meaning that the gravitational force pulling it towards the planet is balanced by its centrifugal force. This can be described by the equation F=mv^2/R, where F is the gravitational force, m is the mass of the satellite, v is its tangential velocity, and R is the distance from the planet.

Now, for the second satellite, it is launched at a speed of 0.5v0, which means that its tangential velocity is only half of that of the first satellite. This means that it will not have enough velocity to balance the gravitational force and will eventually fall towards the planet. The minimum distance between the second satellite and the planet will occur at the point where the gravitational force is at its maximum and the centrifugal force is at its minimum.

To find this minimum distance, we can use the equation for the gravitational force, F=GMm/R^2, and substitute in the values for the second satellite's velocity and the given launch radius R. This will give us the maximum force that the satellite experiences at any point in its orbit. We can then use this force to calculate the minimum distance using the equation F=mv^2/R.

Another approach could be to use the conservation of energy equation, K+U=const, where K is the kinetic energy and U is the potential energy. In this case, the kinetic energy would be 0.5mv^2, and the potential energy would be -GMm/R. Using the given values for the second satellite's velocity and the launch radius, we can solve for the minimum distance r using this equation.

In conclusion, the minimum distance between the second satellite and the planet can be determined by considering the basic principles of gravitation and orbital motion, and using equations such as F=mv^2/R and K+U=const.