Question

a rock thrown vertically upward from the surface of the moon at a velocity of 16 m/sec reaches a height of s = 16t - 0.8t² 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?

Explanation:

Step1: Recall velocity - time formula

The height - time function is $s = 16t-0.8t^{2}$. The velocity function $v(t)$ is the derivative of the position function. Using the power rule $\frac{d}{dt}(at^{n})=nat^{n - 1}$, we have $v(t)=\frac{ds}{dt}=16 - 1.6t$.

Step2: Find time to reach highest - point

At the highest - point, the velocity $v = 0$. Set $v(t)=0$, so $16-1.6t = 0$. Solving for $t$ gives $t=\frac{16}{1.6}=10$ s.

Step3: Find maximum height

Substitute $t = 10$ into the position function $s(t)=16t-0.8t^{2}$. Then $s(10)=16\times10-0.8\times10^{2}=160 - 80=80$ m.

Step4: Find time to reach half - maximum height

First, half of the maximum height is $40$ m. Set $s(t)=40$, so $16t-0.8t^{2}=40$. Rearrange to get $0.8t^{2}-16t + 40 = 0$. Multiply through by 10 to clear the decimal: $8t^{2}-160t + 400 = 0$. Divide by 8: $t^{2}-20t + 50 = 0$. Using the quadratic formula $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ with $a = 1$, $b=-20$, and $c = 50$, we have $t=\frac{20\pm\sqrt{400 - 200}}{2}=\frac{20\pm\sqrt{200}}{2}=\frac{20\pm10\sqrt{2}}{2}=10\pm5\sqrt{2}$ s.

Step5: Find time aloft

The rock is aloft until it returns to the ground, i.e., $s(t)=0$. Set $16t-0.8t^{2}=0$. Factor out $t$: $t(16 - 0.8t)=0$. We have $t = 0$ (initial time) and $16-0.8t=0$, which gives $t = 20$ s.

a. The velocity at time $t$ is $v(t)=16 - 1.6t$ m/s.
b. The time to reach the highest - point:
Set $v(t)=0$, $16-1.6t = 0$, $t = 10$ s.
c. The maximum height:
Substitute $t = 10$ into $s(t)=16t-0.8t^{2}$, $s(10)=80$ m.
d. The time to reach half - maximum height:
Solve $16t-0.8t^{2}=40$, $t = 10\pm5\sqrt{2}$ s.
e. The time aloft:
Solve $16t-0.8t^{2}=0$, $t = 20$ s.

Answer:

a. $v(t)=16 - 1.6t$ m/s
b. $10$ s
c. $80$ m
d. $t = 10\pm5\sqrt{2}$ s
e. $20$ s