Simple Vector Question Related to Conservation of Momentum

[SHISHKABOB]

Homework Statement

A spaceship of mass M is traveling in deep space with velocity v_i = 20 km/s relative to the Sun. It ejects a rear stage of mass 0.2 M with a relative speed u = 5 km/s. What then is the velocity of the spaceship?

Homework Equations

conservation of momentum, p = mv

let U be the velocity of the ejected rear stage and v_f be the velocity of the ship after ejection.

The Attempt at a Solution

p_i = Mv_i
p_f = 0.2MU + 0.8Mv_f

it's an example from the book and the next thing they do is say "The speed u of the ejected stage relative to the spaceship is the difference in velocities of the spaceship and stage"

u = v_f - U

or

U = v_f - ubasically, I am feeling really dumb for not understanding why u = v_f - U. The example also states that the positive direction is in the direction of v_i. If that's so, then shouldn't ube negative? I think I'm thinking about this too hard and have confused myself.

 

[ehild]

Relative velocity with respect to the spaceship can be understood as velocity relative to the final velocity vf. The relative speed of the rear stage with respect to the spaceship is u=5 m/s. The speed is positive, as it is the magnitude of velocity. The relative velocity of the rear stage is opposite to the velocity of the spaceship, so the relative velocity is vr=-u=-5 m/s.
The velocity of the rear stage with respect to the sun is U=vf+vr=vf-u.

ehild