Gravitional potential energy problem

[frostking]

Homework Statement

A space station orbits the sun at the same distance as the Earth but on the opposite side of the sun. A small probe is fired away from the station. What minimum speed does the probe need to escape the solar system?

Homework Equations

Usubg(initial) + Kinitial = Usubg(after probe fired) + K (after)

The Attempt at a Solution

I set - GmM(sun)/ r(sun) + 1/2mv(inital)^2 = 0 + 1/2 mv(after)^2

I then solve for v initial. My problem is that the explanation fro this problem shows that the value of the right side of the equation is 0---that there is 0 potential and 0 kinetic after the probe escapes and I do not understand why, how, etc. the kinetic after is 0! Can someone please tell me why this can be? Thanks very much Frostking

 

[ehild]

To escape from the solar system means that the body has just enough kinetic energy to go to infinity, where its potential energy would be zero. To reach infinity, it has to consume all its kinetic energy, therefore the total energy is also 0 at infinity. As energy is conserved, the total energy of the probe is also zero when it is fired from the station. Do not forget that the potential energy is negative. The gravitational potential at the station is the same as at the position of the Earth.

ehild

 

[frostking]

Thanks very much! It is a bit abstract but I can see the logic of assuming all of the kinetic energy is used up to escape. I will remember that potential energy due to gravity is negative. Thanks again for the assist. Frostking

 

[Relay]

If the probe is launched forward so that it's velocity is added to the space station velocity, it can escape the solar system. If it is launched backwards it will have to reach the station's negative speed and then exceed it. If the station's orbital velocity is increased, it will leave the solar system. If the station's orbital velocity is decreased, it will fall into the sun.
To calculate how long it will take to leave the solar system you will need to know 3 things. The first thing you need to know is the probe velocity relative to the sun. For the second thing you must decide where the solar system ends. The final thing you need to know is the distance of the arc path out of the solar system.
Your minimum speed will depend on when you want the probe to leave the solar system. Just divide the arc distance out by the time to get your velocity. This only works for velocities greater than the station's velocity.