To find the speed of the particle at $t = 6$ seconds, we need to analyze the acceleration vs. time graph and determine the change in velocity over time. The velocity can be found by calculating the area under the acceleration-time graph, as the area represents the change in velocity.

Steps to Solve:

1. Identify the regions under the graph:

   • From $t = 0$ to $t = 4$, the acceleration is constant at $2 \, \text{m/s}^2$.
   • From $t = 4$ to $t = 5$, the acceleration decreases linearly from $2 \, \text{m/s}^2$ to $0 \, \text{m/s}^2$.
   • From $t = 5$ to $t = 6$, the acceleration decreases linearly from $0 \, \text{m/s}^2$ to $-1 \, \text{m/s}^2$.

2. Calculate the area for each region:

   • Area 1 (Rectangle from $0$ to $4$ seconds): $\text{Area}_1 = 2 \times 4 = 8 \, \text{m/s}$

   • Area 2 (Triangle from $4$ to $5$ seconds): $\text{Area}_2 = \frac{1}{2} \times (5 - 4) \times 2 = 1 \, \text{m/s}$

   • Area 3 (Triangle from $5$ to $6$ seconds): $\text{Area}_3 = \frac{1}{2} \times (6 - 5) \times (-1) = -0.5 \, \text{m/s}$

3. Add the areas to find the total change in velocity: $\text{Total change in velocity} = 8 + 1 - 0.5 = 8.5 \, \text{m/s}$

Answer:

The speed of the particle at $t = 6$ seconds is $8.5 \, \text{m/s}$.

Would you like further details or have any questions?

Here are some related questions to explore:

1. How do you find the velocity if the particle had started with an initial velocity?
2. What is the total displacement of the particle during this motion?
3. How would the velocity change if the acceleration graph was inverted?
4. Can we determine the maximum speed reached by the particle from this graph?
5. How would the graph change if there were a constant deceleration instead?

Tip: Remember, the area under an acceleration-time graph always represents the change in velocity.