To determine the speed of the particle, we need to calculate the total distance traveled and then divide it by the total time.

Step 1: Calculate the total distance traveled

The particle's motion involves several steps:

1. From $x = 0.0$ meters to $x = 5.0$ meters: $\text{Distance} = 5.0 - 0.0 = 5.0 \, \text{meters}$
2. From $x = 5.0$ meters to $x = -3.2$ meters: $\text{Distance} = |5.0 - (-3.2)| = 5.0 + 3.2 = 8.2 \, \text{meters}$
3. From $x = -3.2$ meters to $x = 8.0$ meters: $\text{Distance} = |8.0 - (-3.2)| = 8.0 + 3.2 = 11.2 \, \text{meters}$

Now, sum the distances: $\text{Total distance} = 5.0 + 8.2 + 11.2 = 24.4 \, \text{meters}$

Step 2: Convert the total time into seconds

The total time is given as $0.444 \, \text{minutes}$. Since there are 60 seconds in a minute: $\text{Total time} = 0.444 \times 60 = 26.64 \, \text{seconds}$

Step 3: Calculate the speed

The speed is the total distance traveled divided by the total time: $\text{Speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{24.4 \, \text{meters}}{26.64 \, \text{seconds}} \approx 0.916 \, \text{m/s}$

Final Answer:

The speed of the particle is approximately $0.916 \, \text{m/s}$.

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Would you like more details or have any questions about this? Here are some related questions:

1. What is the difference between speed and velocity?
2. How does displacement differ from total distance traveled?
3. Can the average velocity of the particle be calculated?
4. What would the speed be if the time were reduced by half?
5. How does acceleration factor into this problem?

Tip: Speed is a scalar quantity—it only considers magnitude, while velocity includes both magnitude and direction.