Alice stands on Earth's equator. She jumps with vertical velocity $v_0$ and a small horizontal velocity $u$ pointing east, both measured in her own (rotating) frame, reaching a maximum height $h$ that is much less than Earth's radius. She wishes to land back exactly at her launch point on the ground. Earth rotates at angular speed $\omega$, with $\omega R \ll v_0$, and the surface gravity is $g$. Neglect air resistance. What value of $u$ is required?