Let us go back to the original legitimate question, from July 9, that invoked the ancient problem of the “continuum”:
“If you were able to throw an object 'exactly' vertical, such that it came back to you due to gravity, did the object ever 'Stop' at its apex?”,
where Glenn Sprigg is concerned about the change from positive to negative velocity at the apex, and explicitly is not interested in astronomical scales or non-terrestrial situations.
Here, I try to answer that question in three sections: Physics, Engineering, and Quibbling. I start with the velocity directly, since that is more direct than starting with the altitude. Applying this question to pop-up foul balls in baseball could lead us to a “Unified Infield Theory”. (For those who may not be familiar with baseball, that was a joke. Everything else below is serious.)
Physics discussion:
Consider the (idealized) one-dimensional vertical motion of an object in an inertial reference frame with a uniform gravitation field, where the velocity’s vertical component v = v0 > 0 at t = 0, and we define the vertical position y (altitude) at t = 0 to be y = 0 . We do not specify how the object attained that velocity at that position and time, only that, for t > 0, there is no force on the object except gravity. The object goes up (v > 0) to an apex, then falls down (v < 0). The gravitational acceleration (downwards) is -g < 0, where the constant g is positive.
Newtonian physics tells us that, in this model, gravity constantly decelerates the velocity:
v(t) = dy/dt = v0 – g t where v = 0 at a time t = tA = v0 / g , and goes negative thereafter.
Simple calculus tells us that
y(t) = v0 t – (gt2 /2) , where the constant of integration gives y(t) = 0 at t = 0 ,
and that, at time t = tA = v0 / g , the apex altitude is
yA = y(v0 / g) = (1/2) (v02/g) ,
which is a well-known result, but not needed to discuss the velocity. The altitude y is a parabola (quadratic function of time).
We need discuss only the velocity. In the first equation, the velocity is continuous and monotonic, and a simple linear function of time. Looking at the equation for velocity vs. time, we see that it goes continuously through zero.
One could object that the variable does not remain at any value, including zero, for a finite amount of time. However, we define the average velocity at any time t0, over a finite symmetrical time interval 2s, as the mean value from t = (t0 – s) to (t0 + s).
Since the velocity is a linear function of time, the average velocity is independent of the interval length 2s, so long as the interval stays within the time range before the object hits the ground (proof left to the reader). Therefore, the limit of the average velocity as s goes to zero is a trivial computation (since the average velocity is independent of s), equal to v (t) as in the equation above, including the value 0 at the apex (t = tA), where the velocity is positive before the apex and negative thereafter, with an average value of 0 (independent of the interval length).
Engineering discussion:
When higher accuracy is required, engineers modify the above ideal case by adding other details to the calculation, including other factors for the specific problem. Among others, these include:
--Aerodynamics and air resistance. Boeing and General Motors have good software packages for computing these effects. Baseballs have interesting aerodynamic features due to their spin and external stitching. Wrigley Field is famous for winds off Lake Michigan.
--Non-inertial reference frame (earth’s rotation underfoot). Baseball is often played at roughly 40 deg N latitude, and this effect can be added, along with deviations from a true vertical initial velocity.
Quibbles:
Here, I group phenomena that are physically real, but whose effects on the basic question are unmeasureable, especially with respect to uncertainties in the engineering corrections above. Examples include:
--Planck distance. Trivial compared with the 74 mm diameter of a baseball: any effect is not measurable in this problem.
--Non-uniform gravitational field. If the pop-up attains 50 m altitude, the variation in g, with the Earth’s radius of roughly 6400 km, is approximately 16 ppm.
--Special relativity. This is trivial compared with the typical baseball velocity of 45 m/s.
--Heisenberg’s Uncertainty Principle: left as an exercise for the reader.