a rock thrown vertically upward from the surface of the moon at a velocity of 40 m/sec reaches a height of s…

Explanation:

Step1: Recall velocity - derivative of position

Velocity $v$ is the derivative of the position function $s(t)$. Given $s = 40t-0.8t^{2}$, using the power - rule $\frac{d}{dt}(at^{n})=nat^{n - 1}$, we have $v=\frac{ds}{dt}=\frac{d}{dt}(40t-0.8t^{2})=40 - 1.6t$ m/s.

Step2: Recall acceleration - derivative of velocity

Acceleration $a$ is the derivative of the velocity function $v(t)$. Since $v = 40 - 1.6t$, then $a=\frac{dv}{dt}=\frac{d}{dt}(40 - 1.6t)=- 1.6$ m/s². The value of $a = 25$ in the given problem is incorrect.

Answer:

a. Velocity: $v = 40 - 1.6t$ m/s, Acceleration: $a=-1.6$ m/s² b. At the highest - point, $v = 0$. So, $40-1.6t = 0$. Solving for $t$ gives $t=\frac{40}{1.6}=25$ s. c. Substitute $t = 25$ s into the position function $s(t)$. $s(25)=40\times25-0.8\times25^{2}=1000 - 0.8\times625=1000 - 500 = 500$ m. d. First, the maximum height is $s_{max}=500$ m. Half of the maximum height is $s = 250$ m. So, $250=40t-0.8t^{2}$. Rearranging gives $0.8t^{2}-40t + 250 = 0$. Using the quadratic formula $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $ax^{2}+bx + c = 0$ (here $a = 0.8$, $b=-40$, $c = 250$), $t=\frac{40\pm\sqrt{(-40)^{2}-4\times0.8\times250}}{2\times0.8}=\frac{40\pm\sqrt{1600 - 800}}{1.6}=\frac{40\pm\sqrt{800}}{1.6}=\frac{40\pm20\sqrt{2}}{1.6}=\frac{40\pm28.28}{1.6}$. We get $t_1=\frac{40 + 28.28}{1.6}\approx42.7$ s and $t_2=\frac{40 - 28.28}{1.6}\approx7.3$ s. e. The rock is aloft when $s(t)=0$. So, $40t-0.8t^{2}=0$. Factoring out $t$ gives $t(40 - 0.8t)=0$. We have $t = 0$ (initial time) and $40-0.8t=0$, which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which which 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