= Solution
Three experimental tests probe different consequences of <special relativity>. The <Michelson-Morley experiment> compares light round trips along perpendicular interferometer arms while changing the apparatus orientation. Its small observed orientation-dependent fringe shift contradicts the simple stationary-aether prediction and supports orientation-independent round-trip light speed. It does not, alone, exclude every alternative theory.<On the Relative Motion of the Earth and the Luminiferous Ether by Albert Abraham Michelson and Edward W. Morley>[1887 apparatus and fringe-shift comparison]{ref}
In the <muon storage-ring time-dilation experiment>, fast circulating muons live longer in laboratory time than muons at rest. Measuring decay counts versus elapsed laboratory time and independently determining the beam <Lorentz factor> tests $\tau_{\rm lab}=\gamma\tau_0$. Circular motion includes acceleration, so this also tests the applicability of accumulated <proper time> along the stored trajectory.<Measurements of relativistic time dilatation for positive and negative muons in a circular orbit by J. Bailey and others>[storage-ring lifetime measurements and comparison with the relativistic factor]{ref}
<Bertozzi's electron speed-energy experiment> measures <electron> time of flight and deposited <kinetic energy> independently. An electrostatic accelerator and a linear accelerator increase the <energy> while the measured speed approaches the light-speed limit. Calorimetry checks that the additional beam <energy> is real, rather than inferring it only from an assumed voltage-to-speed relation. The result supports relativistic $K=(\gamma-1)mc^2$ rather than extending $K=mv^2/2$ to arbitrarily large speeds.<Speed and Kinetic Energy of Relativistic Electrons by William Bertozzi>[1964 abstract: time-of-flight measurements over 0.5–15 MeV and independent calorimetry]{ref}
For the two ships, set launch time to zero and measure Earth time in years. Their <Lorentz factors> are $\gamma_A=5/4$ and $\gamma_B=5/3$. Bob's thirtieth birthday occurs after one year of his <proper time>, so its Earth coordinates are
$$
t_B=\frac53\,\mathrm{yr},\qquad x_B=-\frac45ct_B=-\frac43c\,\mathrm{yr}.
$$
If Alice sends at Earth time $t_A$, her position is $x_A=3ct_A/5$. The left-going light signal obeys $x=x_A-c(t-t_A)$. Intersection with Bob gives
$$
\frac35t_A-(t_B-t_A)=-\frac45t_B,
\qquad t_A=\frac{t_B}{8}=\frac5{24}\,\mathrm{yr}.
$$
Alice's elapsed <proper time> is therefore
$$
\boxed{\tau_A=\frac{t_A}{\gamma_A}=\frac16\,\mathrm{yr}.}
$$
\b[She sends two months after her twenty-ninth birthday.] This compares the emission and reception events; the captains' clocks need only be synchronized at launch, not kept numerically synchronized afterward.
\Image[/past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1-birthday-signal.png]
{title=A left-going birthday signal intersects Bob's world line one year into his proper time}
{description=Original Earth-frame spacetime calculation with c set to one light-year per year. The emission point is two months into Alice's proper time.}
The relative recession speed is
$$
\frac{|u_{B|A}|}{c}=\frac{4/5+3/5}{1+(4/5)(3/5)}=\frac{35}{37}.
$$
The longitudinal <relativistic Doppler effect> then gives
$$
\boxed{f_B=f_0\sqrt{\frac{1-35/37}{1+35/37}}=\frac{f_0}{6}.}
$$
Equivalently, the left-going <photon> has Earth <frequency> $f_0/[\gamma_A(1+3/5)]=f_0/2$, and Bob's frame supplies another factor $\gamma_B(1-4/5)=1/3$. Both methods give the same redshift.
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