” If the earth speeds up, it will move to a wider orbit around the sun. “
Too fast and not enough for escape velocity only results in a elliptical orbit. Way close then way far away from the sun. Burn & Freeze, burn then freeze.
Even if you tried to it would be difficult to accentuate the ellipse.
As for doing it too fast, that is not going to happen. You would have to pass many thousands of very large comets near the earth to have any noticeable effect. I would be far more worried about getting a calculation wrong or picking the wrong comet or asteroid so that the pull of the earth's orbit puts it into a short term earth crossing orbit. Or worse, causes it to break up and hit the earth like buckshot.
All of this isn't going to happen but it is an interesting academic exercise for mathematicians, engineers and other math literate people.
There are three types of people in the world, those that understand math and those that don't.
Garde la Foi, mes amis! Nous nous sommes les sauveurs de la République! Maintenant et Toujours!
(Keep the Faith, my friends! We are the saviors of the Republic! Now and Forever!)
LonePalm, le Républicain du verre cassé (The Broken Glass Republican)