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Prohibitively expensive.
First the cleanup is gonna take forever and cost billions.
Then building a rocket is gonna be even more billions and time.
And then actually shooting something into the sun is harder than just blasting it out of the solar system.
You could save a bit by shooting it into another star, and not our own. But you still gotta clean it up and make a rocket. I don't think we have even launched a rocket that big or heavy ever. It may require multiple rockets. Planet Express barely was able to make it happen, and they are in the future, only needed to clean NYC, and is also from a cartoon.
This is fun to play with: https://trinket.io/embed/glowscript/6642756b52?toggleCode=true&start=result
Why is this true? Wouldn`t gravity do most of the work if we just kinda shove it in that direction?
Because if you launch something from Earth, you inherit the Earth's orbital speed around the Sun. At that point, whatever you launched will just continue to orbit the Sun. It takes less energy to accelerate to a solar system exit trajectory than it does to scrub off all of the excess velocity and end up on a trajectory that intersects the Sun.
But does it matter what speed the garbage is going at when it hits the sun?
No, but it's going too fast sideways. It would miss the sun. You need to slow it down by the same apeed that Earth is moving, stopping its sideways motion and letting it drop into the sun.
Edit: I like making diagrams. Red is the trajectory you're expecting. Blue is the Earth's motion, which adds to that red arrow. Purple is the resulting actual movement of the trash rocket.
But do you need to slow it down all the way? Can't you just slow it down enough to get the ball in an elliptical orbit where the trash ball gets very close to the ball of plasma?
The problem is slowing it down to any speed that would end up with it dropping into the sun is going to take more effort and be more difficult than firing it out of the solar system. It isn’t practical.
Space is big. It's so big that our tiny ape brains have a hard time conceiving of how big it is. The sun is actually (despite it's size) a relatively small target and is very very far away. Now the more delta-V you burn to slow the trash down the smaller its orbit around the sun will be. But that orbit starts enormous. So to get that purple line near the sun you do need to slow down almost the whole way, just to get it close.
Yes and no. The gravity of the sun will attract the rocket, but there are other things out in space besides the sun.
The problem then is other planets will start whipping the garbage rocket around who knows where. Could even come back around and smash into earth. Same problem with the sun, actually. It's quite hard to hit something that's that big when we're this far away. If you miss even a fraction of a decimal of a degree, the trash rocket will swing around and you're back to planetary hot potato.
It's easier to sling the rocket past the south or north pole at a right angle to the solar plane. Up or down it'll either keep going till it's another suns problem or it joins the Oort cloud, which is kinda like a giant trash dump for everything that didn't make it into our solar system when the sun formed.
Nailed all the points.