Question #43526

A massive spacecraft is moving directly away from Earth with a constant relativistic velocity $V$. Inside the spacecraft, a motorized platform is mounted on a track parallel to the direction of motion. A small light source of mass $m$ is attached to this platform via a spring of force constant $k$. Let $\omega_0 = \sqrt{k/m}$.

Initially ($t \le 0$), the platform is at rest relative to the spacecraft, and the mass is at rest in its equilibrium position.

At $t=0$, the platform begins to oscillate along the track such that its acceleration relative to the spacecraft is $a_p(t) = a_0 \cos(\omega t)$ (where the positive direction is away from Earth), with $\omega = \omega_0 / 2$ and $a_0$ being a positive constant.

The light source continuously emits a signal of proper frequency $\nu_0$ towards Earth. Assume the maximum speed $u$ of the mass relative to the spacecraft satisfies $u \ll c$.

Which of the following expressions gives the maximum frequency of the signal observed on Earth during the subsequent motion?