Question

the height, h (in feet) of a model rocket launched from the roof of a building at t seconds is given by h = s(t)= - 16(t + 2)(t - 6). f. what does the value of c in the formula s(t)=at²+bt + c tell us about the object? g. when does it hit the ground? the object hits the ground at t = 6 sec h. the velocity upon impact is ft per sec

Explanation:

Step1: Expand the height - function

First, expand $h = s(t)=-16(t + 2)(t - 6)$. Using the FOIL method, $(t + 2)(t - 6)=t^{2}-6t+2t - 12=t^{2}-4t - 12$. Then $s(t)=-16(t^{2}-4t - 12)=-16t^{2}+64t + 192$.

Step2: Recall the relationship between position - function and velocity - function

The velocity function $v(t)$ is the derivative of the position function $s(t)$. If $s(t)=at^{2}+bt + c$, then $v(t)=s^\prime(t)=2at + b$.

Step3: Find the velocity function

For $s(t)=-16t^{2}+64t + 192$, where $a=-16$, $b = 64$, and $c = 192$, the velocity function $v(t)=s^\prime(t)=-32t+64$.

Step4: Find the velocity upon impact

The object hits the ground at $t = 6$ seconds. Substitute $t = 6$ into the velocity function $v(t)$.
$v(6)=-32\times6 + 64=-192+64=-128$.

Answer:

$-128$