What kind of a bomb? answer:the exploding kind.

[MathematicalPhysicist]

i have this question from wheeler and taylor book, the answer is that bomb does get explode but i don't know how to explain it obviously T is acclerating so it's not an inertail frame, but still i don't see how will an observer in U see an explosion take place, anyone care to explain, has it got to do with simulatenity?

thnkasin advance.
the pic is in the attachment.

 

[Doc Al]

Realize that an observer in U will see the top of the T smack into the U, while the long arm of the T keeps going. (Since signals cannot travel faster than c, the T cannot be considered a rigid body that stops as a whole when one end is stopped.)

 

[MathematicalPhysicist]

so the answer is there are no rigid bodies, and when the first time the top T smack the U, the T keeps moving with the U so the arms of the U get eventaully shoretend and kaboom, is this correct or am i way off?

thanks in advance.

 

[pervect]

The structural strength of any possible steel should be totally negligible in this problem.
The arms of a bar made out of the strongest steel are not going to be anywhere near strong enough to stop the remainder of the bar. It's doubtful that the U tube will fare very well either. "Deform" is too mild a word to describe what's going to happen to them both when the collision occurs.

The energy contained in the moving bar is probably going to be much greater than the energy in the TNT, as well, considering that 1 ton of TNT has an energy of 4*10^9 joules, while 1 gm of the material in the T has an energy of approx (gamma-1)*10^14 joules, (i.e. (gamma-1)*m*c^2), where 1/gamma is the length contraction factor. We don't have the dimensions, but it's likely that the T is going to weigh a lot more than 1 gm, and in order to get any appreciable length contraction gamma is going to have to be "large", say at least 1.1.So the energy released by the TNT will probably also be negligible, the real fireworks are going to be in the relativistic collision. It's not clear how to model this, and it's probably not the point of the problem. I would guess offhand that you have more than enough energy in the relativistic collision to vaporize the entire assembly. The TNT will probably contribute it's small amount of energy to this process, eventually. However, the detonation speed of the TNT is going to be much slower than the relativistic speeds involved in the collision.

Basically, the relativistic collision will probably result in something that looks a lot like a nuclear fireball. It could easily be a larger energy release than anything in our current nuclear arsenal if the steel 't' bar is not very, very tiny.

 

[MathematicalPhysicist]

it's only a textbook question, you got way off with it pervect.

 

[pervect]

Maybe. I don't really think it is a very good textbook question, though, because if you take it very literally and seriously "as written", you almost have to answer it as I did.

I think that The barn in the pole problem (click on the link for details) is a much better problem along the same lines, that doesn't take you off into tangents if you try to figure out what would actually, physically, happen.

 

[JesseM]

so the answer is there are no rigid bodies, and when the first time the top T smack the U, the T keeps moving with the U so the arms of the U get eventaully shoretend and kaboom, is this correct or am i way off?

thanks in advance.

What do you mean when you say "the arms of the U get eventaully shoretend"? In this problem we assume the U is solid enough that it isn't affected significantly when it's hit by the T, so no matter what frame you pick, the U's length and velocity will remain constant. The point is that the top of the T is stopped when it hits the top of the U, but the bottom of the T can't "know" what happened to the top until a shock wave moving slower than the speed of light travels from the top of the T to the bottom, until that happens the bottom of the T will keep moving along at constant speed. And even if the shock wave traveled at exactly the speed of light it still wouldn't reach the bottom of the T before the bottom of the T hit the bottom of the U (we know this because there is a 'spacelike separation' between the event of the top of the T hitting the top of the U and the event of the bottom of the T hitting the bottom of the U...you can prove there's a spacelike separation by noting that if there's one frame where the T is shorter than the U, and another frame where the U is shorter than the T, then there must be a frame where the T is the same length as the U, and in this frame the two events would be simultaneous...if some pair of events are simultaneous in one frame, they have a spacelike separation in all frames).

 

[JesseM]

Maybe. I don't really think it is a very good textbook question, though, because if you take it very literally and seriously "as written", you almost have to answer it as I did.

It's not actually essential to the problem that the speeds be a significant fraction of c...as long as there's one frame where the length of the T is exactly identical to the length of the U, then in some frames the T will be slightly shorter than the U and in others the U will be slightly shorter than the T, even if their relative speed is only something like 0.0001c or even less.

 

[pervect]

It's not actually essential to the problem that the speeds be a significant fraction of c...as long as there's one frame where the length of the T is exactly identical to the length of the U, then in some frames the T will be slightly shorter than the U and in others the U will be slightly shorter than the T, even if their relative speed is only something like 0.0001c or even less.

I still don't like the problem. The problem says that the T-structure is

made of the strongest steel

It doesn't say that the T-shaped structure doesn't deform (and of course the point is that it must deform).

But if we imagine a T-shaped structure that's actually "made of the strongest steel", it will deform significantly even at .0001c, which is 30 km/sec.

Think of trying to build a steel t-shaped spring which is strong enough to launch itself into orbit at 30 km/sec. I think I can say without doing a full mechanical analysis that that is not possible even "for the strongest steel". It just can't store enough mechanical energy to achive that sort of speed. Offhand, I'd say that you'd be lucky to get some fraction of the speed of sound in steel as an absolute upper limit.

At least at .0001c, the structure will only deform and break, rather than explode in a nuclear fireball :-).

While perhaps I am being a bit silly, it comes from just being very literal, and reading the problem exactly as it was written.

 

[MathematicalPhysicist]

so the long arm of T keeps moving while the shockwave from the back gets to the front up to that event the arm keeps moving and it gets to the TNT, but then it means that the back is deformed while T keeps moving, so the first picture is wrong cause T's shape doesn't stay the same as pervect argued there.
I agree with pervect not intuititve problem.

I have another puzzler which seems quite simple:
a runner has a mirror in his arm, the runner runs near the speed of light will he be able to see himself in the mirror?

my answer is yes, cause in any inertial frame the laws of physics are the same, so if were at rest he would see himself in the mirror i.e the law of reflection still applies also in the inertial frame when he moves compared to the lab for example.