Rocket Escape Velocity from the Earth-Sun system

[Rikudo]

Homework Statement

Earth with mass m orbits the sun (mass M) in circular path (radius R) with velocity v. If a rocket is launched from the earth,find the initial velocity that is needed so that it could leave the solar system (free from both the sun's and earth's gravity).

Note-
this process undergoes two independent steps: free from the earth's gravity, and then free from the sun's gravity.

Relevant Equations

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I have a difficulty when making the energy-conservation-equation for the second step.

When making the equation, we need to know the exact position (measured from the sun) of the rocket after it is freed from the Earth gravitation.

But, where exactly does the rocket free from Earth gravitation? All I know is that this happens at infinity (measured from the earth).

Does this "infinity" negligible if we measure it from the sun? Why?

 

[PeroK]

The Earth's mass is much smaller than the Sun, so you could approximate a long way from Earth as still a distance $R$ from the Sun.

 

[jbriggs444]

Alternately, one could realize that gravitational potential energy is additive. If you add the gravitational potential energy of the rocket in the field of the Earth to the gravitational potential energy of the rocket in the field of the Sun the result is the gravitational potential energy of the rocket in the field of the Sun and Earth combined.

So you can do the calculations in two separate steps and add the results together. But there is no need to do the physical thrusting in two distinct steps. You are actually escaping from both bodies at the same time.

 

[Filip Larsen]

In addition to what has already been mentioned regarding adding two speed delta's being a reasonable approximation, you should also remember to consider that the Earth already moves relative to the Sun.