Question

a body moves on a coordinate line such that it has a position s = f(t)=t² - 8t + 7 on the interval 0 ≤ t ≤ 9, with s in meters and t in seconds.
a. find the bodys displacement and average velocity for the given time interval.
b. find the bodys speed and acceleration at the endpoints of the interval.
c. when, if ever, during the interval does the body change direction?
(type an integer or a simplified fraction.)
the bodys average velocity for the given time interval is 1 m/s.
(type an integer or a simplified fraction.)
the bodys speeds at the left and right endpoints of the interval are 8 m/s and 10 m/s, respectively.
(type integers or simplified fractions.)
the bodys accelerations at the left and right endpoints of the interval are 2 m/s² and 2 m/s², respectively.
(type integers or simplified fractions.)
when, if ever, during the interval does the body change direction? select the correct choice below and fill in any answer boxes within your choice.
a. the body changes direction at t = s.
(type an integer or a simplified fraction.)
b. the body does not change direction during the interval.

Explanation:

Step1: Find displacement

Displacement is $\Delta s=s(9)-s(0)$. Given $s(t)=t^{2}-8t + 7$, then $s(9)=9^{2}-8\times9 + 7=81-72 + 7=16$ and $s(0)=0^{2}-8\times0 + 7=7$. So $\Delta s=16 - 7=9$ meters.

Step2: Calculate average velocity

Average velocity $v_{avg}=\frac{s(9)-s(0)}{9 - 0}=\frac{16 - 7}{9}=1$ m/s.

Step3: Find velocity function

Differentiate $s(t)$ to get velocity $v(t)=s^\prime(t)=2t - 8$.

Step4: Calculate speeds at endpoints

At $t = 0$, $v(0)=2\times0 - 8=- 8$, speed is $|v(0)| = 8$ m/s. At $t = 9$, $v(9)=2\times9 - 8=10$ m/s.

Step5: Find acceleration function

Differentiate $v(t)$ to get acceleration $a(t)=v^\prime(t)=2$ m/s². So $a(0)=a(9)=2$ m/s².

Step6: Determine direction - change time

The body changes direction when $v(t)=0$. Set $2t - 8=0$, then $2t=8$, $t = 4$ s.

Answer:

a. Displacement: 9 m; Average velocity: 1 m/s
b. Speeds at endpoints: 8 m/s, 10 m/s; Accelerations at endpoints: 2 m/s², 2 m/s²
c. A. The body changes direction at $t = 4$ s.