The rockets of the Goths and the Huns are each 1000 m long in their rest frame .

Question

The rockets of the Goths and the Huns are each 1000 m long in their rest frame .The rockets pass each other, virtually touching, at a relativespeed of 0.8c. The Huns have a laser cannon at the rear of theirrocket that shoots a deadly laser beam at right angles to themotion. The captain of the Hun rocket wants to send a threateningmessage to the Goths by "firing a shot across their bow." He tellshis first mate, "The Goths' rocket is length contracted to 600m. Fire the laser cannon atthe instant the nose of our rocket passes the tail of their rocket.The laser beam will cross 400 m in front of them." Butthings are different in the Goths' reference frame. The Gothcaptain muses, "The Huns' rocket is length contracted to 600m 400 mshorter than our rocket. Ifthey fire the laser cannon as their nose passes the tail of ourrocket, the lethal laser blast will go right through o
ur side."
The first mate on the Hun rocket fires as ordered.

Does the laser beam blast the Goths or not?

Resolve this paradox. Show that, when properly analyzed, the Gothsand the Huns agree on the outcome. Your analysis should containboth quantitative calculations and written explanation.

Explanation / Answer

Let the Huns' system be denoted by (H) and the Goths' by (G).
In the Huns' frame, the two events are simultaneous but separated by a distance of 1000 meters, therefore
t(H) = 0
x(H) = 1000 m
We get the corresponding values for the Goths' frame using Lorentz transform,
x(G) = (x(H) - v*t(H)) / (1-(v^2/c^2))
t(G) = (t(H) - v*x(H)/c^2) / (1-(v^2/c^2))
Plugging in for t(H), x(H) from above and v = 0.8c, we obtain
x(G) = (1000 m - 0) / 0.6 = 1667 m
t(G) = (0 - 0.8c * (1000 m) / c^2) / 0.6 = -1333 m / c
Thus, the Goths would observe the laser beam 667 meters ahead of their front, about 4 seconds prior to the contact.
Hense,
the Goths will probably never learn of the laser beam because in the moment it was firing, they were elsewhere. When they come there (moments later), it's already miles away (principle of uncertainty).

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