Note 1: Recall the old relationship "distance = rate x time", so rate (i.e. velocity, or speed) is distance/time. Note that the Galilean transformation states that x' = x - vt, and if both sides of this equation are divided by t, then we have x'/t = x/t - v. This may be written as x/t = x'/t + v, by a simple algebraic operation (solve for x/t). Now, x'/t is just how fast the ball moves as seen from Dick's frame, and since his frame moves with velocity v with respect to Jane, Jane sees the ball's velocity as x/t = x'/t + v. Let's use some numbers: Suppose the train moves at 20 m/s, and Dick throws the ball at 5 m/s. Since Dick and Spot are at rest with respect to the train, they see the ball's speed u' as 5 m/s. Jane sees the ball moving 5 m/s relative to the train, which in turn is traveling in the same direction as the ball, at 20 m/s. In terms of the velocities, u = u' + v, or 25 = 5 + 20. Jane sees the ball moving at 25 m/s.