Answer to Question #147370 in Calculus for Sean

Question

: A toy rocket rises vertically in such a way that t seconds after its liftoff, it is
s(t)= -16t^2 + 200t feet above the ground.
(a) How high is the rocket after 6 seconds?
(b) What is the average velocity of the rocket over the first 6 seconds of flight (between t=0 and t=6)?
(c) What is the instantaneous velocity of the rocket at t=2 sec?

Answer 1

$\displaystyle(a).\\ s(t) = -16t^2 + 200t\\ \begin{aligned} s(6) &= -16(6)^2 + 200(6) \\&= 1200 - 576 = 624\,\textsf{fts} \end{aligned}\\ (b)\\ \textsf{To determine the average velocity}\\ \textsf{of the particle between}\, 1.0 \, s\, \, \textsf{and}\,6.0 \,s,\\ \textsf{we calculate the values of}\\ s(1.0\,s) \,\, \textsf{and}\,\, s(6.0\,s):\\ \begin{aligned} s(6) &= -16t^2 + 200t = -16(6)^2 + 200(6) \\&= 1200 - 576 = 624 \end{aligned}\\ \begin{aligned} s(0) &= -16(0)^2 + 200(0)\\ &= 0 \end{aligned}\\ \begin{aligned} v_{average} &= \frac{s(6) - s(0)}{6 - 0} \\&= \frac{624 - 0}{6 - 0} = \frac{624}{6} \\&= 104 \,\textsf{fts/s} \end{aligned}\\ \textbf{\textsf{OR}}\\ \textsf{Since, velocity is the rate}\\ \textsf{of change of distance with time}\\ v = \frac{\mathrm{d}(s(t))}{\mathrm{d}t} = -32t + 200\\ \begin{aligned} v(6) &= -32(6) + 200\\ &= -192 + 200 = 8 \end{aligned}\\ \begin{aligned} v(0) &= -32(0) + 200\\ &= 200 \end{aligned}\\ \textsf{The sum of the initial and final}\\ \textsf{velocity is divided by}\,2 \, \textsf{to find}\\ \textsf{the average.}\\ \textsf{The average velocity calculator}\\ \textsf{uses the formula that shows the}\\ \textsf{average velocity}\, (v) \, \textsf{equals}\\ \textsf{the sum of the final velocity}\, (v) \, \\ \textsf{and the initial velocity}\, (u), \, \textsf{divided by}\, 2.\\ \begin{aligned} v_{average} &= \frac{v_6 + v_0}{2} \\&= \frac{200 + 8}{2} = \frac{208}{2} \\&= 104 \,\textsf{fts/s} \end{aligned}\\ (c)\\ \begin{aligned} v(2) &= -32(2) + 200\\ &= -64 + 200 = 136\,\textsf{fts/s} \end{aligned}$