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A distance-time function, d(t)
, is in the image. Answer the following questions regarding this d(t)
.

a) What is the average velocity of the entire journey?

b) What is the average velocity between 1-2 seconds into the journey?

c) Of the half-second (0, 0.5, 1, 1.5...) times into the journey, which has the most positive instantaneous velocity?

d) What is the average velocity between 0.5-1 seconds into the journey?

e) Why is the velocity in (d) negative? That is, what does this mean in real-life terms?

f) At what times into the journey is the instantaneous velocity zero?

g) What are the units of average velocity?

h) What are the units of instantaneous velocity?

Let's analyze the function \( d(t) \), which represents distance in meters as a function of time in seconds, and answer the questions you posed.

### a) What is the average velocity of the entire journey?
The average velocity over a time interval is given by:

\[
\text{Average velocity} = \frac{\text{change in distance}}{\text{change in time}} = \frac{d(t_2) - d(t_1)}{t_2 - t_1}
\]

From the graph:
- \( t_1 = 0 \) seconds, \( d(0) = 1 \) meter.
- \( t_2 = 3.5 \) seconds (end of the graph), and \( d(3.5) \approx 10 \) meters.

Thus, the average velocity is:

\[
\frac{d(3.5) - d(0)}{3.5 - 0} = \frac{10 - 1}{3.5 - 0} = \frac{9}{3.5} \approx 2.57 \text{ m/s}.
\]

### b) What is the average velocity between 1-2 seconds into the journey?
From the graph:
- \( d(1) \approx 3 \) meters.
- \( d(2) \approx 6 \) meters.

Average velocity between 1 and 2 seconds:

\[
\frac{d(2) - d(1)}{2 - 1} = \frac{6 - 3}{1} = 3 \text{ m/s}.
\]

### c) Which half-second interval has the most positive instantaneous velocity?
The most positive instantaneous velocity occurs when the curve is steepest. From visual inspection, between 3 and 3.5 seconds, the slope is steepest, indicating the highest positive instantaneous velocity.

### d) What is the average velocity between 0.5-1 seconds into the journey?
From the graph:
- \( d(0.5) \approx 2.8 \) meters.
- \( d(1) \approx 3 \) meters.

The average velocity between 0.5 and 1 seconds:

\[
\frac{d(1) - d(0.5)}{1 - 0.5} = \frac{3 - 2.8}{0.5} = \frac{0.2}{0.5} = 0.4 \text{ m/s}.
\]

### e) Why is the velocity in (d) negative? What does this mean in real-life terms?
If the velocity were negative, it would mean the object is moving backwards, i.e., the distance from the starting point is decreasing. However, in this case, the velocity is positive, though small, indicating a very slight forward motion.

### f) At what times is the instantaneous velocity zero?
Instantaneous velocity is zero where the slope of the curve is flat (horizontal). From the graph, this happens:
- Around \( t \approx 0.75 \) seconds.
- Around \( t \approx 2.5 \) seconds.

### g) What are the units of average velocity?
Average velocity is measured in meters per second (m/s), since it’s the change in distance (meters) divided by the change in time (seconds).

### h) What are the units of instantaneous velocity?
Instantaneous velocity has the same units as average velocity, i.e., meters per second (m/s).

---

Would you like further details on any part of this analysis? Here are some additional questions you might find interesting:

1. How do we calculate instantaneous velocity from a distance-time graph?
2. How does acceleration affect the shape of the distance-time graph?
3. What does the steepness of the curve tell us about the speed?
4. Can you have zero average velocity but non-zero instantaneous velocity? How?
5. How would this graph change if we considered the total distance traveled instead of displacement?

**Tip:** The slope of the distance-time graph gives the velocity, so steeper sections indicate higher speeds!

A distance-time function d(t) is in the image. Answer the following questions regarding this d(t). a) What is the average velocity of the entire journey? b) What is the average velocity between 1-2 seconds into the journey? c) Of the half-second (0, 0.5, 1, 1.5...) times into the journey, which has the most positive instantaneous velocity? d) What is the average velocity between 0.5-1 seconds into the journey? e) Why is the velocity in (d) negative? That is, what does this mean in real-life terms? f) At what times into the journey is the instantaneous velocity zero? g) What are the units of average velocity? h) What are the units of instantaneous velocity?

This solution analyzes a distance-time graph to calculate average and instantaneous velocity. It covers key concepts like the slope of the curve, which represents velocity, and includes answers to questions about time intervals and velocity changes. Suitable for Grades 10-12.

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