I was thinking about asteroids which travel through solar system and I have a question. Is it possible to inject small object into an orbit around one of them and use momentum of this asteroid to increase range of our object?
closed as unclear what you're asking by TildalWave, Hohmannfan♦, Brian Tompsett - 汤莱恩, Rikki-Tikki-Tavi, GdD Feb 12 '16 at 13:22
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This is possible, but entirely pointless. Spacecraft are not limited by range, only by how much they can change their velocity, usually called delta-v. If you can change your velocity enough to catch up with the asteroid, you are already in the same orbit as it. That means, You can do that even if the asteroid was not there.
You are perhaps also thinking about a gravity assist, by which a spacecraft gain momentum from a fly by of a planet. But that only works with massive objects, like planets, and the gain from an asteroid fly-by is incredibly small. Also, if this manoeuvre is performed by first braking into orbit, as you describe, more delta-v is already wasted than you can possibly gain from the manoeuvre.
Orbiting does not work that way.
Getting into orbit around an object means to get close to it and then change your speed and direction to be almost the same as the object. That means you can't "hitchhike" with a fast asteroid, because the energy you would need to rendezvous with it would be the same energy you need to get onto the same trajectory it already has.
What might give you a speed boost is going onto a collision course with the asteroid and having it ram you (like a billard ball). But building a spacecraft which can survive such a collision would be quite challenging.
What might give you a much safer speed boost is doing a fly-by with the asteroid to give you a gravity assist. But the speed-gain by a gravity assist depends on the mass of the object, so you won't get much from an asteroid. A gravity assist is usually only worth the trouble when it's with a planet or at least a large moon.