The goal of the Breakthrough Starshot program (which has $100 million in funding) is to send wafer chip spacecraft to 20% of lightspeed to probe exoplanets and other star systems. The initial $100 million would not reach that goal but a follow on program for a few tens of billions of dollar could achieve it.
Battleship naval guns can fire metal projectiles with 5000Gs of acceleration (49000 meters per second per second).
It would take 367,347 kilometers of distance to accelerate at 5000Gs to reach 2% of light speed.
The moon is 384,400 km away from the Earth.