Classical Mechanics Level 4

The rockets of the Goths and the Huns are each $$1000\text{ m}$$ long in their respective rest frame. The rockets pass each other, virtually touching, at relative speed of $$0.8\text{ c}$$. The Huns have a laser cannon at the rear of their rocket that shoots a deadly laser beam at right angles to the motion. The captain of the Hun rocket wants to send a threatening message to the Goths by “ﬁring a shot across their bow.” He tells his ﬁrst mate, “The Goths rocket is length contracted to $$600\text{ m}$$. Fire the laser cannon at the instant the nose of our rocket passes the tail of their rocket. The laser beam will cross $$400 \text{ m}$$ in front of them.” But things are diﬀerent from the Goths’ perspective. The Goth captain muses, “The Huns’ rocket is length contracted to $$600 \text{ m}, 400 \text{ m}$$ shorter than our rocket. If they ﬁre the laser cannon as their nose passes the tail of our rocket, the lethal laser blast will go right through our side.” The ﬁrst mate on the Hun rocket ﬁres as ordered. Does the laser beam blast the Goths or not? Resolve this paradox.

Note: Assume that the Huns first compute the exact time when the nose of the Hun's rocket will meet with the tail of the Goth's rocket, and this information is relayed to the rear of the Hun's rocket before the Goth's rocket arrives. The exact time is known ahead of time.

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