Galilean transformations

Inertial observer on ground throws ball in air (blue) with initial horizontal velocity vox.
Moving observer in rail car (with initial velocity vc and acceleration ac) observes ball's motion (red).

ac=0, vox=0, vc=1

ac=0, vox=1, vc=1

ac=0, vox=0, vc=-1

ac=0, vox=0, vc=-2

ac=2, vox=0, vc=-2

ac=4, vox=0, vc=-2

ac=6, vox=0, vc=-2

ac=8, vox=0, vc=-2