= Solution
{c}
Let the object be at rest in $S'$, with <proper length> $L_0=x'_2-x'_1$. Measuring its length in $S$ requires two endpoint <events> simultaneous in $S$, so $\Delta t=0$. The <Lorentz transformation> then gives
$$
L_0=\Delta x'=\gamma(\Delta x-v\Delta t)=\gamma L,
\qquad
\boxed{L=\frac{L_0}{\gamma}=L_0\sqrt{1-v^2/c^2}.}
$$
This is <length contraction>. The simultaneous endpoint <events> in $S$ are not simultaneous in $S'$; that <relativity of simultaneity> is essential to the measurement.
For the two rockets, take the first rocket's Earth-frame <velocity> as $+c/2$ and the second's as $-c/2$. The <velocity-addition formula> gives the second rocket's <velocity> in the first rocket's <rest frame>:
$$
u=\frac{-c/2-c/2}{1-(-c/2)(c/2)/c^2}=-\frac45c.
$$
Its <Lorentz factor> is $\gamma_{\mathrm{rel}}=5/3$, and therefore its measured length is
$$
\boxed{L_{\mathrm{other}}=\frac35L_0.}
$$
Here “appear” is interpreted as a length measured using simultaneous endpoint positions in the observer's <inertial frame>. A photograph, involving unequal light travel times, is a different measurement.
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