A spacecraft of proper length (its length in its own frame ) moves at along the…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Special relativity: spacetime and kinematics · Special relativity: spacetime and kinematics

Relativity and Gravitation, Ch Time dilation, length contraction and simultaneity

Problem

A spacecraft of proper length

30.0\ \mathrm{m}

(its length in its own frame

S'

) moves at

v = 0.6c

along the common

+x

axis of the station frame

S

. Both frames are inertial and in standard configuration, with clocks synchronised within each frame. Asked for the length of the craft in the station frame, a student writes: "The inverse transformation is

\Delta x = \gamma(\Delta x' + v\,\Delta t')

. I mark the two ends of the craft at the same instant, so

\Delta t' = 0

and

\Delta x = \gamma\,\Delta x' = 1.25\times 30.0\ \mathrm{m} = 37.5\ \mathrm{m}

. The craft is

37.5\ \mathrm{m}

long in the station frame." Take

c = 2.998\times 10^{8}\ \mathrm{m\,s^{-1}}

. ((a)) Identify the error in one or two sentences, naming the frame in which the end events must be simultaneous. ((b)) Give the correct length of the craft in the station frame. ((c)) Find the time interval in the station frame between the pair of events the student actually used, to show that the station crew could not have used them as a length measurement.

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