Let's work through the questions based on the velocity vs. time graph you uploaded:

a. At what two times is the object at rest? An object is at rest when its velocity is zero. From the graph, the velocity is zero at two points:

• At $t = 0$ s (where the velocity crosses the zero line on the positive side).
• At $t = 8$ s (where the velocity crosses the zero line on the negative side).

b. At what times is the object not accelerating? The object is not accelerating when the velocity is constant. From the graph, we can see that the velocity is constant between:

• $t = 4$ s to $t = 8$ s.
• $t = 12$ s to $t = 16$ s.

c. What is the acceleration at $t = 4$ s? The acceleration is the slope of the velocity vs. time graph. At $t = 4$ s, the graph shows a sharp change, indicating a sudden acceleration. Specifically, between $t = 4$ and $t = 8$ s, the slope is constant (indicating constant acceleration). The slope is negative, meaning negative acceleration (deceleration).

d. What is the acceleration at $t = 16$ s? At $t = 16$ s, the velocity is constant again (horizontal line from $t = 12$ to $t = 16$). This means the acceleration at $t = 16$ s is zero.

e. At what two times is the acceleration undefined? The acceleration is undefined where there is an instantaneous change in velocity. These occur at:

• $t = 0$ s, where the velocity jumps from 0 to a positive value instantly.
• $t = 12$ s, where the velocity jumps from a negative value to 0 instantly.

f. At what times is the acceleration defined and changing? The acceleration is defined (i.e., has a specific value) and changing between:

• $t = 8$ s and $t = 12$ s, where the velocity is changing in a non-linear way.
• The slope is not constant here, meaning the acceleration is changing.

g. What is the instantaneous speed at $t = 10$ s? At $t = 10$ s, the graph shows a negative velocity of about -4 m/s, so the instantaneous speed is $4 \, \text{m/s}$.

h. For what values is the object’s instantaneous speed increasing? The object’s speed is increasing when the magnitude of the velocity is increasing. This happens when the velocity is moving away from zero (either in the positive or negative direction). From the graph:

• Between $t = 0$ s and $t = 4$ s, the speed is increasing as the velocity goes from 0 to 12 m/s.
• Between $t = 12$ s and $t = 16$ s, the speed increases from -8 m/s to 0 m/s.

Would you like any more clarification on these points? Feel free to ask!

Here are some related questions to explore further:

1. How does the shape of the graph relate to the object’s acceleration?
2. What is the significance of constant velocity segments in motion analysis?
3. How does the change in direction impact the acceleration and speed of an object?
4. Can we use the area under the velocity vs. time graph to find displacement?
5. How does the concept of instantaneous velocity differ from average velocity?

Tip: Remember, acceleration is the slope of a velocity-time graph!