Yet another spaceship moving at .9c thread

[pervect]

Yet another "spaceship moving at .9c" thread

What I'd like to see a thread about is what would happen if a spaceship, moving at .9c, were to impact a planet. Let's say the spaceship was 100,000 metric tons, roughly the size of a Nimitz-class aircraft carrier.

It's not hard to figure out the energy of the spaceship, but it's harder to estimate the physical effects.

 

[robphy]

As a first approach, model the Earth and the ship as two particles.
Assuming a totally-inelastic collision, calculate the energy and momentum transferred.

Some facts via Google
"mass of Earth = 5.9742 × 10^24 kilograms"
and
"1 metric ton = 1 000 kilograms"
and
"1 / sqrt(1 - (.9^2)) = 2.29415734".

 

[HallsofIvy]

I assume you mean the REST MASS of the spaceship.. As robphy pointed out, the "relativistic factor" at .9 c is only 2.29 so you are now talking "what if 2 and a quarter ships crashed into a planet. Doubt if it would do a whole lot!

 

[russ_watters]

2.29 ships at .9C is still a lot of kinetic energy though...

 

[Fredrik]

Wow. I'd like to know the answer to that question too. The kinetic energy of the ship is

$$(\gamma-1)mc^2$$

where m is the rest mass, and since gamma is roughly 2 in this case, the kinetic energy is close to the rest mass energy of the ship.

Since m=10^8 kg and c is roughly 3*10^8 m/s, this means that the kinetic energy is about 10^25 J.

That's a lot of Joules.

 

[russ_watters]

2.29 ships at .9C is still a lot of kinetic energy though...

If a typical meteor hits the atmosphere at 30,000mph, an object of the same mass going .9c has 400 million times the kinetic energy. Thats considerably more energy than all the worlds' nuclear weapons put together. Anyone have a link to that impact calculator website (lemme see if I can find it...)?

 

[russ_watters]

HERE it is. I'm not sure if I'm doing this right, but it gives me a 10^9 megaton air-burst with a shock wave that at 1000 miles is 27 psi and flattens pretty much everything in its path.

 

[Fredrik]

10^25 Joules is of order 10^9 megatons (according to onlineconversion.com), so it seems to be correct so far.

 

[pervect]

Just to clarify, I mean the invariant mass of the ship, i.e. the rest mass, is 100,000 metric tons (this is about the same as the displacement mass of the Nimitz class carriers).

When I run the figures through the URL for the impact calculator, I use

(62 meter radius sphere of 100 kg/m^3 density -> 124 meter diameter, 100 kg/m^3 density)

and I get

Energy before atmospheric entry: NaN x 10NaN Joules = 8.68 x 10^8 MegaTons TNT

(which is .8 petatons, peta=10^15)

But I don't think that's right. I get 2*10^25 joules for the total energy of the ship. 2e25 joules is 4.7 petatons of TNT according to

online conversion

not .8 petatons.

 

[Haelfix]

Dunno but its likely the impact equation is not using the correct relativistic laws, since its highly unnusual in practise anyway.

Either way, the point is that its a monumental explosion, that pretty much levels the Earth's surface and all life pretty damn fast. (Kinda puts some perspective on the alien crash claims) assuming say the size of an aircraft carrier.

 

[franznietzsche]

yeah relativistic velocities would not occur for something actually striking the earth.

Unless we really pissed off an alien race who then loaded a nimitz class carrier into a giant rail gun at shot it at us at .9c. So long as Bush is not president at first contact we should be ok...

 

[pervect]

I rather laboriously computed the gravitational binding energy of a sphere of constant density of radius R, only to find the result with a websearch

here

It turns out that the gravitational binding energy of the Earth is 2.24*10^32 joules. The impact calculated earlier in the thread has 2*10^25 joules of KE, so it has about one ten-millionth of the binding energy of the Earth.

The binding energy of the Earth's atmosphere is 3*10^26 joules (by my calculation, -G Mearth Matmos / Rearth, with Matmos = 5*10^21 kg.) So the impact has about 1/15 of the gravitational binding energy of the atmosphere.