Question

a rock thrown vertically upward from the surface of the moon at a velocity of 8 m/sec reaches a height of $s = 8t - 0.8t^2$ meters in t sec.
a. find the rocks velocity and acceleration at time t.
b. how long does it take the rock to reach its highest point?
c. how high does the rock go?
d. how long does it take the rock to reach half its maximum height?
e. how long is the rock aloft?

a. find the rocks velocity at time t.
$v = \square$ m/s

Explanation:

Step1: Differentiate height for velocity

Velocity $v(t)$ is the derivative of height $s(t)=8t-0.8t^2$. Using power rule: $\frac{d}{dt}(t^n)=nt^{n-1}$.
$v(t)=\frac{d}{dt}(8t-0.8t^2)=8 - 1.6t$

Step2: Differentiate velocity for acceleration

Acceleration $a(t)$ is the derivative of velocity $v(t)=8-1.6t$.
$a(t)=\frac{d}{dt}(8-1.6t)=-1.6$

Step3: Find time at highest point

At highest point, $v(t)=0$. Solve $8-1.6t=0$.
$1.6t=8 \implies t=\frac{8}{1.6}=5$

Step4: Calculate maximum height

Substitute $t=5$ into $s(t)$.
$s(5)=8(5)-0.8(5)^2=40-0.8(25)=40-20=20$

Step5: Find time for half max height

Half max height is $\frac{20}{2}=10$. Solve $8t-0.8t^2=10$.
Rearrange: $0.8t^2-8t+10=0$. Multiply by 10: $8t^2-80t+100=0$. Divide by 4: $2t^2-20t+25=0$. Use quadratic formula $t=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ where $a=2, b=-20, c=25$.
$t=\frac{20\pm\sqrt{400-200}}{4}=\frac{20\pm\sqrt{200}}{4}=\frac{20\pm10\sqrt{2}}{4}=\frac{10\pm5\sqrt{2}}{2}$
Simplify: $t=5-\frac{5\sqrt{2}}{2}\approx1.46$ (ascending) and $t=5+\frac{5\sqrt{2}}{2}\approx8.54$ (descending)

Step6: Find time aloft

Rock is aloft when $s(t)=0$. Solve $8t-0.8t^2=0$.
$t(8-0.8t)=0$. Solutions: $t=0$ (launch) and $8-0.8t=0 \implies t=\frac{8}{0.8}=10$

Step1: Differentiate height function

Velocity is derivative of $s(t)=8t-0.8t^2$.
$v(t)=\frac{d}{dt}(8t-0.8t^2)=8-1.6t$

Answer:

a. Velocity: $v = 8 - 1.6t$ m/s; Acceleration: $a = -1.6$ m/s²
b. $5$ sec
c. $20$ meters
d. $t=5-\frac{5\sqrt{2}}{2}$ sec (ascending) and $t=5+\frac{5\sqrt{2}}{2}$ sec (descending)
e. $10$ sec

For the specific fill-in part (a, velocity only):