The Wikipedia article on Gravity assist has this terrestrial analogy:
Imagine standing on a train platform, and throwing a tennis ball at 30 km/h toward a train approaching at 50 km/h. The driver of the train sees the ball approaching at 80 km/h and then departing at 80 km/h after the ball bounces elastically off the front of the train. Because of the train's motion, however, that departure is at 130 km/h relative to the train platform; the ball has added twice the train's velocity to its own.
I have two problems with this:
The train's speed relative to the platform is 50 km/h. The ball, traveling in the opposite direction relative to the same platform, is going at 30 km/h. Thus, the train operator will indeed see the ball at
50 + 30 = 80 km/h.
So far, so good. However, once the ball has hit the front of the train (let's assume that no speed at all is lost in the process), it'll have the train's velocity (50 km/h) added to its own (30 km/h) relative to the platform:
50 + 30 = 80 km/h.
Because the train's velocity relative to the platform is 50 km/h, and the ball's now 80 km/h, the operator will see it departing from him at
80 - 50 = 30 km/h.
As already mentioned above, the ball's velocity relative to the platform after the bounce will be:
The train's velocity (50 km/h) + the ball's initial velocity (30 km/h) = 80 km/h, and not twice the train's velocity.
What am I missing?