Related to this question. A lot of the discussion there got sidetracked from the question of orbital physics, into questions about alien motives and what the humans would do, so I want to start a new question with more clear physical parameters and a more well-defined objective.
You have a motor that can exert 100N of force over a very long period of time (at least a thousand years). It may be an ion drive, powered by a large solar panel array, and capable of refining extra propellant from comets and asteroids. It's 1AD, you begin in Earth's orbit, and you want to do as much damage to humanity as you can, by smashing asteroids or comets into Earth with as much kinetic energy as possible. If you can actually kill humanity, that is your ideal outcome. Your spaceship weighs 10000 kg. You can easily compute trajectories to the desired precision. What methods do you use?
Some possible strategies:
- "The drop": take a very massive asteroid or an Oort cloud object and slow it down until it drops out of orbit. Just let it fall to Earth while you move on to the next asteroid.
- "The windup": take a less massive Oort cloud object and ferry it even further out of the solar system, then slow it to a stop when at a sufficient distance. Then accelerate it straight at Earth.
- Suicide run: like "the windup," but without an asteroid. Just get very far from the solar system, then come to a stop, turn around and accelerate straight at Earth. For this purpose we may say the spaceship weighs 10000kg. If the approach phase of the suicide run takes 1000 years, the impact energy will be around 10^21 J, without taking into account relativity. (Chicxulub = 10^25 J). Can be done only once.
- Gravity assist: drive your object past Jupiter so that it slingshots around it, gaining speed.
- Multiple gravity assist: repeat the gravity assist several times on one object.
- NEO wrangling: take an existing large comet or asteroid on a highly eccentric orbit that will pass near Earth, such as Halley's comet or 99942 Apophis. Catch it far from Earth, then adjust its orbit slightly so it hits Earth. This sounds like a good plan to me, although it is limited by the number of such NEOs.
- Simultaneous targeted attack: set up dozens of objects precisely targeting cities, timed to strike in rapid succession and wipe out all the major cities on Earth.
- Moon tugboat: take a small moon and drive it towards Earth. (sounds not feasible)
- One big hit causing an extinction event. Chicxulub-scale energy (10^25 J) would definitely do it, if you can get there.
- Multiple Tambora-sized hits (10^20 J) to affect the climate globally. One would need a lot of these in rapid succession to cool the planet enough to cause extinctions.
- Many smaller nuke-sized hits to wipe out cities (such as 10^15 J).
If you're still around past AD 2000, we can assume humans will be causing problems for you, unless you've already done substantial damage or are already moving too fast to be stopped. AD 2500 without doing damage = Mars is colonized, game over.
Please no answers without a back-of-the-envelope calculation of feasibility, the impact energy, and time required. See TNT equivalent for some examples of different energetic events for comparison. Also wolfram alpha is great for unit conversions.