“There will be peace when the people of the world, want it so badly, that their governments will have no choice but to give it to them. I just wish you could all see the Earth the way that I see it. Because when you really look at it, it’s just one world.” -Superman
So, you want to shoot Earth’s garbage into the Sun, do you? From a physics point-of-view, this is difficult, but possible. First, you have to overcome the pull of Earth’s gravity, and escape into space. Next, you have to take into account the fact that Earth orbits the Sun at 30 km/s, and to fall into the Sun, you need to take that all away, something that would require a series of gravity assists.
NASA’s flight path for the Messenger probe, which wound up in a successful, stable orbit around Mercury after a number of gravity assists. Image credit: NASA / JHUAPL, via http://messenger.jhuapl.edu/About/Mission-Design.html.
But even if you figure all the physics out — and even if cost is no object to you — you need to figure out how many launches it would take and how low you can get your launch failure rate. Even if we got the failure rate down to just 0.1%, more than an order of magnitude lower than it’s ever been, we’d still spread hundreds of thousands of points of radioactive waste all over Earth.