Quick back-of-the-envelope calculation:
Approx. mass of Pacific Ocean[0]: m_ocean = 7.1×10²⁰kg
Specific heat of water: c = 4.2kJ/(kg · K)
Temperature of Pacific Ocean: T_1 ~ 293K
Temperature at which water starts boiling: T_2 ~373K
=> Energy needed to make Pacific Ocean boil:
E_heat = c m_ocean ΔT = c m_ocean (T_2 - T_1) ~ 3×10²⁶ J
On the other hand, the relativistic kinetic energy formula is:
E_kin = (γ-1) m c²,
where γ = 1/sqrt(1-v²/c²) = 1/sqrt(1-0.95²) and m is the space junk's mass.
Setting E_kin = E_heat therefore yields:
=> m = E_heat / [(γ-1)c²) = 3×10²⁶ J / (2.2×10¹⁶ m²/s²)] = 10¹⁰ kg
For comparison: The mass of all of humanity combined is somewhere between 10¹¹kg and 10¹²kg. Now those numbers do look somewhat comparable but:
- We haven't taken into account the space ships required to transport everyone
- E_heat was waste heat but since practically all energy will become waste heat at the end of the day, E_heat gives us a pretty good estimate of the total energy we will have (had) access to.
All in all 0.95·c doesn't seem feasible for moving humanity to Proxima Centauri, given E_heat. For moving 10¹⁰ kg of space junk, sure, though I'm not sure what you were planning to do with all that space junk in the first place?
[0]: https://en.wikipedia.org/wiki/Pacific_Ocean
