There are two identical uniform spherical planets of radius R. The first has its center at the origin of the xyz coordinate system. The second has its center at (2R, 0, 0). The planets are touching.

A projectile is launched from the "North Pole" of the first planet at (0, 0, R) with its initial velocity pointed in the direction of the vector (1, 0, 1).

Let the escape speed relative to the planet's surface be v_e. Note that here, the escape escape is for a single planet in isolation (following the typical convention).

With the given launch vector, let v₀ be the minimum launch speed for the projectile to reach the "North Pole" of the second planet at (2R, 0, R).

How are the two speeds v_e and v₀ related?