|MadSci Network: Physics|
There are two previous answers in our archives that touch on this questioin:
The argument can be made that the object spends no time at all at the apogee, based on arguments similar to the previous answers'. Prior to apogee the upward velocity continuously decreases until the upward velocity becomes zero. At that exact instant the downward velocity is also zero, and from then on the downward velocity increases. The "instant" consists of zero time!!
John Link, MadSci Physicist
Try the links in the MadSci Library for more information on Physics.