Question

the accompanying figure shows the velocity v = \\(\frac{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.
choose the correct graph of the acceleration, where defined.

Explanation:

Step1: Recall direction - change condition

The body reverses direction when velocity changes sign. From the graph, velocity $v = f(t)$ changes sign at $t = 3$ s and $t = 6$ s.

Step2: Recall constant - speed condition

The body moves at a constant speed when the magnitude of velocity is constant. This occurs when the graph of $v(t)$ is a horizontal line. So, it is moving at a constant speed on the intervals $[5,7]$ s.

Step3: Graph - speed

Speed is the absolute - value of velocity. For $0\leq t\lt3$, $s(t)=v(t)$; for $3\leq t\lt5$, $s(t)= - v(t)$; for $5\leq t\lt7$, $s(t)=v(t)$; for $7\leq t\lt10$, $s(t)= - v(t)$.

Step4: Recall acceleration formula

Acceleration $a=\frac{dv}{dt}$. The acceleration is the slope of the velocity - time graph. For $0\lt t\lt3$, $a>0$; for $3\lt t\lt5$, $a < 0$; for $5\lt t\lt7$, $a = 0$; for $7\lt t\lt10$, $a<0$. The correct graph of acceleration is the one that has positive values when the velocity is increasing, negative values when the velocity is decreasing, and zero values when the velocity is constant.

Answer:

a. The body reverses direction at $t = 3$ s and $t = 6$ s.
b. It is moving at a constant speed on the interval $[5,7]$ s.
c. (Graph of speed needs to be drawn based on the absolute - value of the given velocity graph).
d. (Based on the slope of the velocity graph, the correct graph of acceleration is the one that matches the description of positive, negative, and zero slopes as per the velocity - change intervals. Without seeing the full - scale options clearly, we can't choose from A, B, C, D precisely, but the acceleration is positive when velocity is increasing, negative when velocity is decreasing and zero when velocity is constant).