I want to know how fast the space shuttle would have to hit the earth to unleash the same kinetic energy as the asteroid that killed the dinosaurs. I've already determined that it's relativistic, by showing that Newtonian physics gives a v greater than c

Spoiler: Newtonian physics

4.2e25J = (2e6 * v ^ 2 ) / 2.
8.4e25J = 2e6 * v ^ 2
v = sqrt(8.4e25J / 2e6)
v = 6,480,740,698 m/s

But once I plug in the equations, I get a wrong result.

KE = (m * c ^ 2) / sqrt(1 - (v ^ 2 / c ^ 2))
KE / (m * c ^ 2) = sqrt(1 - (v ^ 2 / c ^ 2))
((KE / (m * c ^ 2)) ^ 2) - 1 = -(v ^ 2 / c ^ 2)
v = -sqrt((((KE / (m * c ^ 2)) ^ 2) - 1) * c ^ 2)

KE = 4.2e25J
c = 1
m = 2e6kg

v = -sqrt((((4.2e25 / (2e6)) ^ 2) - 1) * 1)
v = -2.1e+19

I'm expecting an answer between 0 and 1, where did I go wrong?

Where? I divided both sides of the equation by the term m * c ^ 2? It was in the numerator of the RHS in the 1st line and in the denominator of the LHS on the second line. I'm sure this is going to be one of those things that's stupidly obvious once I see it, but I'm still missing it.

I did? Isn't that the value I'm supposed to use? If I want v as a percentage of c, shouldn't c be 1 exactly?

There's a few other problems to deal with in this scenario: they had big problems with re-entry even at very low velocities. Like most meteorites, your speeding shuttle would evaporate in the first few km of atmosphere -- fortunately.

##v \approx c##, and ##c \approx 3 \times 10^8 m/s##, so if the atmosphere is 100km thick, the shuttle will take approximately 0.3 milliseconds to traverse it. Will the spaceship evaporate in that period? Does it matter? You're still depositing a mole of joules into the system.

Awesome, that's the exact same answer I got when I put in 299792458 instead of 3e5, just with an i in it. I'm sure I'd find the mistake if I looked, but I don't really care if my mistake only changes the imaginary factor. Thanks for the simplification too.

I was thinking that too. I doesn't really make a difference, you're still adding the amount of energy that killed the dinosaurs into the system. It'd go off like a bomb and obliterate the biosphere no matter where something with that much power exploded.

Granted. I got stuck in the immediate vicinity of the actual speeds of descent and forget to 'think' relativistically.
Does it matter ? No. There's no way you can get this kind of energy in a 2000 metric ton vehicle. Think of it: the energy equivalent of 470 kilotonne of matter ...

But it sure is fun as a thought experiment... and to bicker about