To determine the particle's speed at $t = 10.0 \, \text{s}$, we need to find the area under the acceleration-time graph from $t = 0 \, \text{s}$ to $t = 10.0 \, \text{s}$. The area under the curve in an acceleration-time graph gives the change in velocity (speed, in this case, because the particle starts from rest and the direction of acceleration is not specified).

The graph consists of simple geometric shapes, so we can break the time intervals into parts where the acceleration is constant and compute the areas.

Steps:

1. From $t = 0 \, \text{s}$ to $t = 5 \, \text{s}$:

   • Acceleration, $a_x = 2 \, \text{m/s}^2$.
   • The area of this section is a rectangle with width $5 \, \text{s}$ and height $2 \, \text{m/s}^2$: $\text{Area} = 5 \, \text{s} \times 2 \, \text{m/s}^2 = 10 \, \text{m/s}.$

2. From $t = 5 \, \text{s}$ to $t = 10 \, \text{s}$:

   • Acceleration, $a_x = 0 \, \text{m/s}^2$.
   • The area is zero, as there is no acceleration: $\text{Area} = 0 \, \text{m/s}.$

Total velocity at $t = 10 \, \text{s}$:

• The total change in velocity is the sum of the areas from $0 \, \text{s}$ to $10 \, \text{s}$. $v_{10} = 10 \, \text{m/s}.$ Thus, the particle's speed at $t = 10.0 \, \text{s}$ is 10 m/s.

Would you like further details or clarifications? Here are some related questions you might explore:

1. How would you calculate the particle's velocity at $t = 15 \, \text{s}$?
2. What is the particle’s final velocity at $t = 20 \, \text{s}$?
3. How would you calculate the displacement of the particle over the time interval $t = 0 \, \text{s}$ to $t = 20 \, \text{s}$?
4. What is the significance of the negative acceleration between $t = 15 \, \text{s}$ and $t = 20 \, \text{s}$?
5. How does the particle's motion change when the acceleration is zero?

Tip: Always remember that the area under an acceleration-time graph corresponds to the change in velocity.