Question

an object moves in a straight line with position after t hours given by s(t)=2t^3 - 24t + 20, t≥0. here distance is measured in miles. a) is it moving forwards (ie direction of increasing s) or backwards when t = 1? what about when t = 5? does it ever stop? b) what is the acceleration when t = 1 and t = 5? is it speeding up or slowing down at these times?

Explanation:

Step1: Find the velocity function

The velocity function $v(t)$ is the derivative of the position - function $s(t)$. Using the power rule $\frac{d}{dt}(t^n)=nt^{n - 1}$, if $s(t)=2t^{3}-24t + 20$, then $v(t)=s^\prime(t)=6t^{2}-24$.

Step2: Determine the direction of motion at $t = 1$

Substitute $t = 1$ into $v(t)$: $v(1)=6\times1^{2}-24=6 - 24=-18$. Since $v(1)<0$, the object is moving backwards at $t = 1$.

Step3: Determine the direction of motion at $t = 5$

Substitute $t = 5$ into $v(t)$: $v(5)=6\times5^{2}-24=6\times25-24 = 150 - 24=126$. Since $v(5)>0$, the object is moving forwards at $t = 5$.

Step4: Find when the object stops

The object stops when $v(t)=0$. Set $v(t)=6t^{2}-24 = 0$. Then $6t^{2}=24$, $t^{2}=4$, and $t = 2$ (we consider $t\geq0$).

Step5: Find the acceleration function

The acceleration function $a(t)$ is the derivative of the velocity - function $v(t)$. Since $v(t)=6t^{2}-24$, then $a(t)=v^\prime(t)=12t$.

Step6: Find the acceleration at $t = 1$ and $t = 5$

When $t = 1$, $a(1)=12\times1 = 12$. When $t = 5$, $a(5)=12\times5=60$.

Step7: Determine if the object is speeding up or slowing down

The object is speeding up when $v(t)$ and $a(t)$ have the same sign, and slowing down when they have opposite signs.
At $t = 1$, $v(1)=-18<0$ and $a(1)=12>0$, so the object is slowing down at $t = 1$.
At $t = 5$, $v(5)=126>0$ and $a(5)=60>0$, so the object is speeding up at $t = 5$.

Answer:

a) At $t = 1$, the object is moving backwards. At $t = 5$, the object is moving forwards. The object stops at $t = 2$.
b) The acceleration at $t = 1$ is $12$ miles per hour squared and the object is slowing down. The acceleration at $t = 5$ is $60$ miles per hour squared and the object is speeding up.