# Thread: The Derivative as a Rate of Change

1. ## The Derivative as a Rate of Change

A rock thrown vertically upward from the surface of the moon at a velocity of $\displaystyle 24$ $\displaystyle \frac{m}{sec}$ reaches a height of $\displaystyle s=24t-0.8t^2$ meters in $\displaystyle t$ $\displaystyle sec$.
a. Find the rock's velocity and acceleration at time $\displaystyle t$ $\displaystyle \rightarrow v=24-1.6t$ $\displaystyle \frac{m}{s}$ $\displaystyle a=-1.6$ $\displaystyle \frac{m}{s^2}$
b. How long does it take the rock to reach its highest point? $\displaystyle \rightarrow 24-1.6t=0$ $\displaystyle t=15$ $\displaystyle sec$
c. How high does the rock go? $\displaystyle \rightarrow s_{max}=24(15)-0.8(15)^2$ $\displaystyle s_{max}=180$ $\displaystyle m$
d. How long does it take the rock to reach half its maximum height?
e. How long is the rock aloft? $\displaystyle \rightarrow 30$ $\displaystyle sec$

I don't know how to find how long it takes the rock to reach half its max height. How would I solve this?

2. Originally Posted by VitaX
I don't know how to find how long it takes the rock to reach half its max height. How would I solve this?
Well, once ya find the max height (call it $\displaystyle s_{max}$) then throw $\displaystyle \frac{s_{max}}{2}$ into the left side of your displacement function and then solve for $\displaystyle t$.

3. Originally Posted by VonNemo19
Well, once ya find the max height (call it $\displaystyle s_{max}$) then throw $\displaystyle \frac{s_{max}}{2}$ into the left side of your displacement function and then solve for $\displaystyle t$.

overlooked that easy bit. So when I put them in the quadratic formula, do I disregard one answer or take both as when its going up and then going down?

4. Originally Posted by VitaX
overlooked that easy bit. So when I put them in the quadratic formula, do I disregard one answer or take both as when its going up and then going down?
Good question. I thought about that when I answered your post. I would tend to say that the question is concerned about the upward motion, but I wouldn't neglect the downward.

I would use both, but I would explain my answer to avoid confusion and cover my bases.