How fast would a rocket have to be going in order to completely leave the solar system? (For the second part of this problem, assume that the Sun is the only mass in the solar system and that the rocket is beginning its journey from Earth's orbit?)
It must have enough acceleration to overcome the net gravitational force of the solar system.
Second part. The rocket will just need enough acceleration to overcome the gravitational force of the sun and the earth. Well, if the earth was moving in orbit you can use centripital force to also launch your rocket outwards, which would mean that you would not need to counter the earth`s gravitational force(since you`re using it for acceleration). Then you can also use the sun`s gravity for centripital force.... Too much theory, hopefully that's not grade-school work.