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 - 0.8t² meters in t sec. a. find the rock’s 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 rock’s velocity at time t v = □ m/s

Explanation:

Step1: Recall velocity - displacement relation

Velocity $v$ is the derivative of displacement $s$ with respect to time $t$. Given $s = 8t-0.8t^{2}$, by the power - rule of differentiation $\frac{d}{dt}(at^{n})=nat^{n - 1}$, we have $v=\frac{ds}{dt}$.
$v=\frac{d}{dt}(8t - 0.8t^{2})=8-1.6t$

Step2: Recall acceleration - velocity relation

Acceleration $a$ is the derivative of velocity $v$ with respect to time $t$. Since $v = 8-1.6t$, then $a=\frac{dv}{dt}$.
$a=\frac{d}{dt}(8 - 1.6t)=- 1.6$

Step3: Find time to reach highest point

At the highest point, the velocity $v = 0$. Set $v=8 - 1.6t = 0$.
$8-1.6t=0$, then $1.6t = 8$, and $t=\frac{8}{1.6}=5$ s

Step4: Find maximum height

Substitute $t = 5$ s into the height formula $s = 8t-0.8t^{2}$.
$s=8\times5-0.8\times5^{2}=40 - 20=20$ m

Step5: Find time to reach half - maximum height

Half of the maximum height is $s = 10$ m. Substitute $s = 10$ into $s = 8t-0.8t^{2}$, we get the quadratic equation $0.8t^{2}-8t + 10=0$. Multiply through by 10 to get $8t^{2}-80t + 100 = 0$, or $2t^{2}-20t + 25=0$.
Using the quadratic formula $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $ax^{2}+bx + c = 0$. Here $a = 2$, $b=-20$, $c = 25$.
$t=\frac{20\pm\sqrt{(-20)^{2}-4\times2\times25}}{2\times2}=\frac{20\pm\sqrt{400 - 200}}{4}=\frac{20\pm\sqrt{200}}{4}=\frac{20\pm10\sqrt{2}}{4}=5\pm\frac{5\sqrt{2}}{2}$

Step6: Find time the rock is aloft

The rock is aloft when $s = 0$. Set $s=8t-0.8t^{2}=0$. Factor out $t$: $t(8 - 0.8t)=0$.
We have two solutions: $t = 0$ (corresponds to the time of throwing) and $8-0.8t=0$, which gives $t = 10$ s

Answer:

a. $v = 8-1.6t$ m/s
b. $t = 5$ s
c. $s = 20$ m
d. $t=5\pm\frac{5\sqrt{2}}{2}$ s
e. $t = 10$ s