a body moves on a coordinate line such that it has a position s = f(t) = \\frac{16}{t^{2}}-\\frac{4}{t} on…
a body moves on a coordinate line such that it has a position s = f(t) = \\frac{16}{t^{2}}-\\frac{4}{t} on the interval 1 ≤ t ≤ 4, with s in meters and t in seconds.\na. find the bodys displacement and average velocity for the given time interval.\nb. find the bodys speed and acceleration at the endpoints of the interval.\nc. when, if ever, during the interval does the body change direction?\nthe bodys displacement for the given time interval is m.\n(type an integer or a simplified fraction.)
Answer
Explanation:
Step1: Recall displacement formula
Displacement $\Delta s=s(4)-s(1)$.
Step2: Calculate $s(4)$
Substitute $t = 4$ into $s(t)=\frac{16}{t^{2}}-\frac{4}{t}$. [ \begin{align*} s(4)&=\frac{16}{4^{2}}-\frac{4}{4}\ &=\frac{16}{16}- 1\ &=1 - 1\ &=0 \end{align*} ]
Step3: Calculate $s(1)$
Substitute $t = 1$ into $s(t)=\frac{16}{t^{2}}-\frac{4}{t}$. [ \begin{align*} s(1)&=\frac{16}{1^{2}}-\frac{4}{1}\ &=16-4\ &=12 \end{align*} ]
Step4: Calculate displacement
[ \begin{align*} \Delta s&=s(4)-s(1)\ &=0 - 12\ &=- 12 \end{align*} ]
Answer:
$-12$