Question

the accompanying figure shows the velocity v = ds/dt = f(t) (m/sec) of a body moving along a coordinate line.
a. when does the body reverse direction?
b. when is it moving at a constant speed?
c. graph the bodys speed for 0≤t≤10.
d. graph the acceleration, where defined.

Explanation:

Step1: Identify direction - change points

The body reverses direction when the velocity changes sign. From the graph, this occurs at $t = 5$ and $t = 8$.

Step2: Identify constant - speed intervals

A constant speed means the magnitude of velocity is constant. From the graph, this occurs on the intervals $[1,3]$ and $[3,5]$.

Step3: Graph speed

Speed is the absolute - value of velocity. For $0\leq t\leq1$, $v(t)$ is increasing from $0$ to $2$. For $1\leq t\leq3$, $v(t)=2$. For $3\leq t\leq5$, $v(t) = 2$. For $5\leq t\leq6$, $v(t)$ is decreasing from $2$ to $- 2$. For $6\leq t\leq8$, $v(t)$ is increasing from $-2$ to $2$. For $8\leq t\leq10$, $v(t)$ is decreasing from $2$ to $0$.

Step4: Graph acceleration

Acceleration $a(t)=\frac{dv}{dt}$. On $[0,1]$, $a(t)$ is a positive constant (slope of $v(t)$). On $[1,3]$, $a(t) = 0$. On $[3,5]$, $a(t)=0$. On $[5,6]$, $a(t)$ is a negative constant. On $[6,8]$, $a(t)$ is a positive constant. On $[8,10]$, $a(t)$ is a negative constant.

Answer:

a. $t = 5,8$
b. $1\leq t\leq3,3\leq t\leq5$
c. Graph speed as described above (absolute - value of the given velocity graph).
d. Graph acceleration as described above (slope of the velocity graph).