How fast and in what direction with respect to the Earth’s orbit should the rocket be fired?

Question

The orbital velocity of the Earth about the sun is 30 km/s. If the Earth were suddenly stopped in its tracks, it would simply fall radially into the Sun. Devise a plan whereby a rocket loaded with radioactive wastes could be fired into the Sun for permanent disposal. How fast and in what direction with respect to the Earth’s orbit should the rocket be fired?

Answer 1

You could fire the rocket 30 km/s opposite Earth's motion around the sun, but you'd save a lot of fuel if you took advantage of the slingshot effect to slow the rocket as it passes the moon and again when it passes planets, including Earth several times. Whatever method you use, it will probably cost about as much energy as what you got from the material before it became waste. If that's the case, it would have been less costly to have simply left the uranium ore in the ground to begin with.