It seems that major league baseball players have to wait a relatively long time to field a pop-up. This led me to wonder how high the ball traveled. If one knew how high the ball traveled, one could calculate speed and time approximately.
Not sure if you're looking for an "average" popup or exceptional ones.
A website with some facts on Tropicana Field says the height of the "A" ring catwalks ranges from 181 feet in center to 194 feet near home plate. A Cleveland.com piece has similar figures, and also suggests that as of 2008 (representing play over 18 seasons), no one had hit the catwalks at that height.
A New York Times article states that two players had hit the "A" ring as of 2010, including Jason Kubel at "about 190 feet" high. Balls could (rarely) be hit higher than this in other locations and wouldn't be noted. So I'd suggest an upper limit to flies at around 200 feet.
So, if we make a few assumptions, we can get a reasonable estimate.
First, I'm going to take the time from contact to when it's fielded at 6 seconds -- I got this by taking the video of Hosmer dropping the Bregman popup, which took 24 seconds at 1/4 playback speed; i.e., 6 seconds of real time.
Next, I'm going to disregard air resistance to simplify the physics. From here, we can invoke two elementary physics formulas:
t = (2*v*sin(theta)) / g
h = (v^2*sin(theta)^2) / (2*g)
v is the exit velocity off the bat,
theta is the launch angle, and
g is the gravitational acceleration (9.8 m/s^2).
Rearranging the first equation for v gives
v = g*t / (2*sin(theta)). Now, we can plug that into the second equation. Doing so (after some algebra) gives
h = g*t^2 / 8. Very conveniently, we don't need to know (estimate) the exit velocity or the launch angle. Now, we simply insert our values, which gives
h = (9.8 m/s^2) * (6 s)^2 / 8 = 44.1 m, which is roughly 145 feet. Hence, we can conclude that a popup can go roughly 145 feet high.
Mike Laga hit a pop fly out of the old Busch Stadium.
Estimated at around 250 feet high and over 300 in the air to make it completely over the roof. Not sure I have ever seen a ball hit so high in my life.