But the golf ball at the other end isn't pushed out at exactly the same time. That signal propagates at the speed of sound through the golf balls, which is actually quite slow.
But here's the correct answer, chapter and and verse being in section 7.11 of Classical Electrodynamics by J.D. Jackson (2nd edition, p. 319):
The qualitative features of the propagation of a signal are now clear. Some minute part of the wave propagates with the velocity of light in vacuum. This initial signal, called the first or Sommerfeld precursor, is very small and oscillates rapidly. At a later time t1 there is a sudden change when omega=0 becomes a point of stationary phase. The second or Brillouin precursor, of greater amplitude and longer period, arrives. At later times, depending on the details of n(omega) and the incident wave, the signal settles down to the expected steady-state behavior. It is evident that the exact build up of the signal is a complicated matter, that causality and relativity are obeyed regardless of the detailed dispersive properties of the medium, and that the arrival of the signal cannot be given an unambiguous definition. The general usage is to take the group velocity of the dominant frequency component as the signal velocity and velocity of energy transport. This suffices in most circumstances, but with sensitive enough detectors the signal velocity can evidently be pushed close to the velocity of light in vacuum, independent of medium.
Noting like the words of a master. The only thing I can add to this wonderful passage is that recently, techniques have been developed for "decapitating" the precursor and discarding the rest of the signal, resulting in effective phase velocities that are greater than c. Despite the lurid headlines this past summer, this does not mean that information can be transmitted faster than c.
Imagine, if you will a long train leaving Baltimore for DC at 30 mph, where the train is as long as the distance between the cities. The long train officially leaves Baltimore when the center of the train leaves Baltimore. Unfortunately, just as the front of the train is about to arrive in DC, the first few cars (the Sommerfeld precursor) break away and the rest of the train gets stuck. The short train officially gets to DC when its center arrives in DC. The calculated speed of the train is 60 mph, even though no car in the train ever went faster than 30 mph.