Question

1. -/4 points details my notes ask your teacher practice another scalcet9 3.1.053. the equation of motion of a particle is s = t^3 - 12t, where s is measured in meters and t is in seconds. (assume t ≥ 0.) (a) find the velocity and acceleration as functions of t. v(t)= a(t)= (b) find the acceleration, in m/s^2, after 4 seconds. m/s^2 (c) find the acceleration, in m/s^2, when the velocity is 0. m/s^2 g(r)=√r + ^8√r g(r)= g(r)=

Explanation:

Step1: Rewrite the function $G(r)$

Rewrite $G(r)=\sqrt{r}+\sqrt[8]{r}$ as $G(r)=r^{\frac{1}{2}}+r^{\frac{1}{8}}$.

Step2: Find the first - derivative $G'(r)$

Using the power rule $\frac{d}{dr}(r^n)=nr^{n - 1}$, we have $G'(r)=\frac{1}{2}r^{\frac{1}{2}-1}+\frac{1}{8}r^{\frac{1}{8}-1}=\frac{1}{2}r^{-\frac{1}{2}}+\frac{1}{8}r^{-\frac{7}{8}}$.

Step3: Find the second - derivative $G''(r)$

Differentiate $G'(r)$ again using the power rule. $G''(r)=\frac{1}{2}\times(-\frac{1}{2})r^{-\frac{1}{2}-1}+\frac{1}{8}\times(-\frac{7}{8})r^{-\frac{7}{8}-1}=-\frac{1}{4}r^{-\frac{3}{2}}-\frac{7}{64}r^{-\frac{15}{8}}$.

For the motion problem:

Step1: Find the velocity function $v(t)$

Since velocity $v(t)$ is the derivative of the position function $s(t)$, and $s(t)=t^{3}-12t$. Using the power rule $\frac{d}{dt}(t^n)=nt^{n - 1}$, we get $v(t)=\frac{d}{dt}(t^{3}-12t)=3t^{2}-12$.

Step2: Find the acceleration function $a(t)$

Since acceleration $a(t)$ is the derivative of the velocity function $v(t)$. Differentiating $v(t)=3t^{2}-12$ with the power rule, we have $a(t)=\frac{d}{dt}(3t^{2}-12)=6t$.

Step3: Find the acceleration at $t = 4$

Substitute $t = 4$ into the acceleration function $a(t)$. $a(4)=6\times4 = 24$ m/s².

Step4: Find when the velocity is 0

Set $v(t)=0$, so $3t^{2}-12 = 0$. Add 12 to both sides: $3t^{2}=12$, then divide by 3: $t^{2}=4$, which gives $t = 2$ (since $t\geq0$).

Step5: Find the acceleration when the velocity is 0

Substitute $t = 2$ into the acceleration function $a(t)$. $a(2)=6\times2=12$ m/s².

Answer:

$G'(r)=\frac{1}{2}r^{-\frac{1}{2}}+\frac{1}{8}r^{-\frac{7}{8}}$
$G''(r)=-\frac{1}{4}r^{-\frac{3}{2}}-\frac{7}{64}r^{-\frac{15}{8}}$
$v(t)=3t^{2}-12$
$a(t)=6t$
$24$
$12$