Themed Challenge (Frozen): The Evergrowing Olaf

There are two unknowns in the following, the initial speed, $x$ , and the time taken for the snowball to reach the apex, $t$ . Since the final velocity of the snowball is $0$ ,

$0$ $=$ $x^{2}$ + $2\times-9.8\times s$ , where $s$ is the displacement of the snowball, to the apex.

Olaf is $0.5$ meters tall, and grows at a rate of $10$ $cm./s$ . Thus, the height of Olaf at a time $t$ is given by $0.5+0.1\times t$

Thus, $s$ $<$ $0.5+ 0.1\times t$

Thus, plugging in, $x^{2}$ $<$ $9.8 + 1.96\times t$

Now, since final velocity is $0$ ,

$0$ $=$ $x$ $-9.8\times t$

Thus, $x$ $=$ $9.8\times t$

Plugging in $9.8^{2}t^{2}$ $<$ $9.8 + 1.96t$

The maximum value of $t$ is approximately $0.33$

Thus, $x$ $=$ $9.8(0.33)$ , which rounds down to $\boxed{3}$