The problem is asking us to find the moment in time $t$ when the velocity of the material point is equal to $2$, given the function $y = S'(t)$ shown in the graph (which is the derivative of the position function $S(t)$).

Analysis:

1. The graph of $S'(t)$ is given, and it represents the velocity of the point.
2. To find the time $t$ when the velocity $S'(t) = 2$, we simply look at the graph and determine the $t$-value(s) for which $y = 2$.

Observation:

From the graph:

• The line $y = 2$ intersects the curve of $S'(t)$ at $t = 1 \, \text{sec}$ (as read from the $x$-axis).

Conclusion:

The velocity of the point equals $2$ at $t = 1 \, \text{sec}$.

Would you like a detailed step-by-step explanation of interpreting the graph or additional context about derivatives? Let me know!

Here are 5 related questions to deepen your understanding:

1. What is the significance of $S'(t)$ in terms of motion?
2. How would you find the total distance traveled by the material point on $[0, 8]$?
3. If the graph of $S'(t)$ represents velocity, what would the graph of $S(t)$ look like?
4. How would the graph change if the material point were accelerating instead of moving at constant velocity?
5. Could there be more than one point where $S'(t) = 2$? Why or why not?

Tip: Always remember that the derivative of a position function $S(t)$ gives you the velocity, while the second derivative gives the acceleration!