If you and your friend both push on the bike at the same time, with the same strength, but in opposite directions, what happens to the motion of the bicycle
Nothing happens to the speed of the bicycle relative to the ground.. it continues to travel at 10 mph easterly since the equal and opposite reaction of you and your friend balance on the bicycle. Only the speed and position of the jumpers changes (with respect to the ground)
If you and your friend push off southerly and northerly. then you are also still moving at 10 mph to the east so your vectored speed is angular to the northeast and southeast.. no effect on the bicycle
If you and your friend push of easterly and westerly, then the relationship of the bicycle to the person going westerly and the person going easterly is unchanged unless you count the impact of the bicycle with the easterly projected person who would be run over by the bicycle when he/she comes to a stop but the bicycle (which you said has a constant speed and direction) catches up to him/her.
