What amount of force would it take to overcome the inertia of the earth's movement in space relative to the Su

Question

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Answer 1

1 micronewton of force would be enough, given enough time.

You need to review exactly what you mean by things like force, impulse, and momentum.

Answer 2

It's the total energy one needs to know to "knock" the planet Earth out of the solar system, not force, because whether or not a force would succeed in "overcoming the inertia of the earth's movement" depends on how long it's applied. The escape velocity for the sun at the vicinity of the Earth's orbit is 42.1 km/sec, while the Earth's orbital speed is 29.8 km/sec. The mass of the Earth is 5.97 x 10^24 kg. The energy required to do the job is:

E = (1/2)(42.1^2 - 29.8^2)(10^6)(5.97)(10^24) = 2.64 x 10^33 Joules

If matter were to be converted totally into energy as per E=mc^2, something like all the water in Lake Baikal, the largest fresh water lake in the world at 23,000 km^3, would have to be converted into energy in order to knock the Earth out of the solar system.