I recently calculated using $E = 0.5mv^2$ that one kilogram of mass traveling at one tenth of the speed of light would have a kinetic energy of 449.4 trillion joules. Relativistic effects are insignificant at this speed. This is the amount of energy equivalent to an earthquake of 7 on the Richter scale or a large hydrogen bomb.
This is an enormous amount of energy from the perspective of designing a propulsion system to accelerate this 1 kg space craft to this speed from rest even if it is started from space outside of earth's gravity. Conventional chemical rockets would be out of the question and much slower accelerations using alternate propulsion systems would be needed and even then present a very significant challenge to eventually obtain said velocity.
We know that Voyager 1 and 2 have used gravity assist flybys of planets Jupiter, Saturn, Uranus, and Neptune to eventually increase their velocities to 17000m/sec for Voyager 1 and 15000m/sec for Voyager 2. While this is fine escaping the solar system and enter deep space, it is still very slow for reaching Proxima Centauri, the closest star group after our sun, in any practical time. Even at one tenth the speed of light, 29,979,246 m/sec, this would take 40 years. 40 years is still a long time but a practical and manageable long term goal.
I was wondering if a great many chained gravity assists within our solar system could be used to eventually accelerate the 1 kg space craft to said speed. This article explains gravity assists well, and how they can be used to speed up or slow down spacecraft relative to the solar frame of reference.
I have in mind oscillating gravity assists on planets in orbits on opposite sides of the sun, back and forth to eventually reach this speed. I realize that the spacecraft can only gain speed if it the exit trajectory of the gravity assist is in the forward orbital tangent from the planet. This would present a challenge if we would like to keep oscillating the gravity assists from planet to planet, but it doesn't seem impossible.