We make the assumptions that \(-0.5 \lt h \lt 0.5\) and \(h \ne 0\) because \(h\) cannot be zero (otherwise there is no interval on which to compute average velocity) and because the function only makes sense on the time interval \(0 \le t \le 1\text{,}\) as this is the duration of time during which the ball is falling. Observe that we want to compute and simplify

\begin{equation*}
AV_{[0.5, 0.5+h]} = \frac{s(0.5+h) - s(0.5)}{(0.5+h) - 0.5}\text{.}
\end{equation*}

The most unusual part of this computation is finding \(s(0.5+h)\text{.}\) To do so, we follow the rule that defines the function \(s\text{.}\) In particular, since \(s(t) = 16-16t^2\text{,}\) we see that

\begin{align*}
s(0.5+h) \amp = 16 - 16(0.5 + h)^2\\
\amp = 16 - 16(0.25 + h + h^2)\\
\amp = 16 - 4 - 16h - 16h^2\\
\amp = 12 - 16h - 16h^2\text{.}
\end{align*}

Now, returning to our computation of the average velocity, we find that

\begin{align*}
AV_{[0.5, 0.5+h]} \amp = \frac{s(0.5+h) - s(0.5)}{(0.5+h) - 0.5}\\
\amp = \frac{(12 - 16h - 16h^2) - (16 - 16(0.5)^2)}{0.5 + h - 0.5}\\
\amp = \frac{12 - 16h - 16h^2 - 12}{h}\\
\amp = \frac{-16h - 16h^2}{h}\text{.}
\end{align*}

At this point, we note two things: first, the expression for average velocity clearly depends on \(h\text{,}\) which it must, since as \(h\) changes the average velocity will change. Further, we note that since \(h\) can never equal zero, we may further simplify the most recent expression. Removing the common factor of \(h\) from the numerator and denominator, it follows that

\begin{equation*}
AV_{[0.5, 0.5+h]} = -16 - 16h\text{.}
\end{equation*}

Now, for any small positive or negative value of \(h\text{,}\) we can compute the average velocity. For instance, to obtain the average velocity on \([0.5,0.75]\text{,}\) we let \(h = 0.25\text{,}\) and the average velocity is \(-16 - 16(0.25) = -20\) ft/sec. To get the average velocity on \([0.4, 0.5]\text{,}\) we let \(h = -0.1\text{,}\) which tells us the average velocity is \(-16 - 16(-0.1) = -14.4\) ft/sec. Moreover, we can even explore what happens to \(AV_{[0.5, 0.5+h]}\) as \(h\) gets closer and closer to zero. As \(h\) approaches zero, \(-16h\) will also approach zero, and thus it appears that the instantaneous velocity of the ball at \(t = 0.5\) should be \(-16\) ft/sec.