Question

One hundred years in the future, waste from nuclear power plants has built up to unmanageable quantities. In an effort to get rid of the stuff, a mission is being planned to remove the waste from the Earth in a spacecraft and chuck it into space. Setting aside the fact that it’s probably a really stupid idea to put a large amount of highly dangerous stuff on top of a big column of explosives, the question arises as to where the craft should be sent once it leaves the Earth. Two options are presented: 1) send it into the Sun, or 2) send it out of the solar system. The craft starts out in the Earth’s orbit around the Sun, but is free of the Earth’s gravity. If no planetary fly-bys are permitted, and the craft is to remain in the Earth’s orbital plane, which option requires the least energy expenditure?

Answer 1

Earth is moving around the sun with about 30km/s. To leave the solar system we need to reach the escape velocity of sun which is about 42km/s. As Earth is already moving with 30km/s, so the probe just need to gain additional 42-30 = 12km/s in order to leave the solar system.

However if the probe needs to dive in the sun it has to stop all of its motion in order to dive in the Sun. To do that the probe needs to go to the opposite direction of Earth's movement with Earth's velocity, i.e. 30km/s. This is more than twice (2.5) the initial velocity of the spacecraft if it would have left the solar system.

As Energy is proportional to the square of the velocity, so it'll take 2.5² = 6.2 times more energy to send the probe to the sun than to send it outside the solar system.

Option (2) is correct.