Re: Not sure of the math
REALLY not sure of the math, but what would we be looking at regarding the destructive power of a 100Kg projectile moving at 0.3C?
Relativistic projectiles begin approaching the yield of an equal mass of matter/anti-matter. At 0.3c, plug it into the kinetic energy equation: E = 0.5 x 100kg x (30% light speed in m/s)^2 = 4E17 Joules. With a bit of rounding, that's 100 megatons of TNT.
Note that if you were aiming this at Earth from a LaGrange point (as suggested in another post), you wouldn't have enough distance to boost the projectile to 0.3c. 15,000G's can cross 400,000km in 73.77 seconds. (Huh, I was expecting a bit longer.) The projectile will only reach 0.036c in that time.
It might be a viable asteroid defense, though I'd be curious to see how a c-fractional projectile interacts with a rubble pile asteroid. At 0.3c and 100 megatons per 100kg, you've kind of left the realm of merely kinetic projectiles behind and start behaving like a 100kg blob of atomic particles when they pass into the asteroid. Energy transfer would also occur quite differently than in an equivalent nuclear bomb. You'd probably end up with most of the energy transfer occurring at some depth into the hide of the asteroid like a near-surface or underground nuke, which would turn most asteroids smaller than a Dino-Killer into a killer cloud of gravel.
The laser itself is a more versatile tool for dealing with asteroids. Direct vaporization wouldn't be completely feasible for large (1km+) asteroids without months of adiabatic heating by a ~100GW laser, and adiabatic heating won't occur in a glowing-hot asteroid. But there's a lot you could do with such a big laser.
You can gently pressure the asteroid at lower power and low power densities, avoiding catastrophic off gassing or break ups. Or you can crank it up and burn small spots to generate a plume and thus propulsion, limited by the integrity of the asteroid. You can deal with individual fragments or even illuminate an entire cloud of gravel. And you can spend days and months working on an asteroid.
The amount of photon propulsion from such a big laser is nice but if you have a lot of mass you don't mind vaporizing you can greatly crank up the thrust available. 100GW might deliver ~200lbs of thrust for a photon sail, but if you're vaporizing rock and water at ~1,000m/s exhaust velocity (Specific impulse of 100) you could get, like, 204 million pounds of thrust. I don't think that estimate if off by more than an order of magnitude.