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Say I want to shoot a cannonball into the Sun with minimal energy (minimal initial velocity relative to Earth).
In which direction do I shoot it?

Let's neglect Earth's gravity, if that would make things very complicated.

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Goes nicely with Staying in orbit - but doesn't any perturbation start a positive feedback?. With a little thought you should be able to deduce the method from the answer there. – dmckee Feb 11 '13 at 20:03
The actual minimal energy trajectory is probably a gravity slingshot off some other convenient nearby body (e.g. moon, mars and/or venus), or maybe a low energy transfer using the "interplanetary transport network". – Ilmari Karonen Feb 11 '13 at 21:19

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In user20847's scenario you would have to fire the cannonball at about 30 km/s to counteract the earth's orbital speed. That is a lot of energy.

Here's a alternative that would use less energy. Launch the ball at the correct time and speed so it starts to fall towards the sun but comes close enough to Venus for its path to be deflected and directed towards the sun. You could use Mercury for a second deflection and use even less energy. More complicated paths with multiple fly-bys are also possible.

This method is called "gravity assist" and is used regularly by NASA to reduce the energy required for sending spacecraft to other planets. For example, the Messenger spacecraft that is in orbit around Mercury received two boosts from Venus and one from Earth - but not in that order! Details of its path are here.

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If we only neglect earth's gravity, then you still have the gravity of the sun. Then you have a rotating ball alround the sun. What you are saying is the same as if you say there is no earth, only a cannonball or another model would be: Which how much energy would you 'shoot' earth, so that it leaves the orbit and falls into the sun.

If you make it easy, you assume the sun is a dot. Then just shoot into the opposite direction moving direction of earth with just the same velocity as earth has around the sun. The earth will have zero speed an fall into the sun. If it does not have zero speed it cannot exactly hit the 'dot'-sun.

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