Finally, I understand how gravity assists for spacecraft work now! (scroll to bottom of linked page)
Imagine a ball rolling down a hill. It gains speed rolling downhill, but then loses speed as it rolls up the next upslope. It’s hard to see how speed can be permanently gained this way. But now imagine that the hill is being propelled forward as you roll down it. Now you’re not only gaining speed due to the slope, but due to the motion of the hill as well…