A liner of proper length leaves rest at the origin of an inertial frame at and burns…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry · The equivalence principle and gravity as geometry

Relativity and Gravitation, Ch Accelerated observers, hyperbolic motion and the equivalence principle

Problem

A liner of proper length

60\ \mathrm{m}

leaves rest at the origin of an inertial frame

S

t = \tau = 0

and burns along the common

+x

axis in two stages, always in the same direction. In the first stage its accelerometer reads

a_1 = 9.8\ \mathrm{m\,s^{-2}}

for a proper time

\tau_1 = 1.5\times 10^{7}\ \mathrm{s}

; in the second it reads

a_2 = 4.9\ \mathrm{m\,s^{-2}}

for a further proper time

\tau_2 = 2.0\times 10^{7}\ \mathrm{s}

. Nose and tail may be given the same proper acceleration to the accuracy wanted here. ((a)) Find the liner's speed measured in

S

at the end of the second stage, and the Lorentz factor

\gamma

of the liner in

S

there. ((b)) Clocks sit at the nose and at the tail. Working to first order in

a h/c^{2}

, find how much time the nose clock gains on the tail clock during the second stage alone. ((c)) During the second stage the crew feel a steady

4.9\ \mathrm{m\,s^{-2}}

and say they might as well be resting on the surface of a spherical planet of radius

4.0\times 10^{6}\ \mathrm{m}

whose surface field strength is

4.9\ \mathrm{m\,s^{-2}}

. In a sealed capsule falling freely just above that planet's surface, two marbles are released from rest one directly above the other,

2.0\ \mathrm{m}

apart. Find their relative acceleration just after release, and the time for the gap to grow by

10\ \mu\mathrm{m}

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