Why Does Time Dilation Not Apply When Rocket A's and Earth's Tails Meet?

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Time dilation does not apply uniformly when two relativistic rockets meet because the events of their noses and tails meeting occur at different positions in the earthbound frame. When rocket A and an earthbound observer set x=t=0 at the moment their noses meet, the time for the tails to meet, T, is measured differently in each frame. The time dilation formula, which applies to a single moving clock, cannot be directly applied to the scenario of two separate events occurring at different locations. The key point is that the two events do not share the same spatial coordinates in the earthbound frame, leading to discrepancies in the time recorded. Thus, the time experienced by rocket A for the tails meeting is not simply \gammaT.
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Suppose two relativistic rockets A and B are headed towards each other. When the noses meet, rocket A and an earthbound observer set x=t=0. Let T be the time, relative to the earthbound observer, that the tails of the rockets meet. Why is it not the case that the time that the tails meet relative to rocket A is \gammaT ?
 
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That time dilation formula applies to a single moving clock. Do the two events (noses meet; tails meet) occur at the same position in the earthbound frame (and thus the time difference is recorded by a single earthbound clock)? Not necessarily.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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