To solve this problem, we will break it down into two phases of the sprinter's motion:

1. Phase 1 (Acceleration phase):
   The sprinter accelerates from rest for $t_1 = 3.0 \, \text{s}$ with a constant acceleration $a_x$. The distance covered during this phase will be denoted as $d_1$.

2. Phase 2 (Constant speed phase):
   After accelerating, the sprinter runs with constant speed for the remaining time $t_2 = 12.0 \, \text{s} - 3.0 \, \text{s} = 9.0 \, \text{s}$. The distance covered during this phase will be denoted as $d_2$.

The total distance is 100 m, so $d_1 + d_2 = 100 \, \text{m}$.

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Phase 1: Accelerating for 3 seconds

The distance covered during the acceleration phase is given by the equation of motion: $d_1 = \frac{1}{2} a_x t_1^2$ The velocity at the end of this phase, $v_1$, is: $v_1 = a_x t_1$

Phase 2: Constant speed for 9 seconds

During this phase, the sprinter moves with constant speed $v_1$ for $t_2 = 9.0 \, \text{s}$. The distance covered in this phase is: $d_2 = v_1 t_2 = a_x t_1 t_2$

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Total distance:

The total distance is the sum of $d_1$ and $d_2$: $d_1 + d_2 = 100$ Substituting the expressions for $d_1$ and $d_2$: $\frac{1}{2} a_x t_1^2 + a_x t_1 t_2 = 100$ Now, substituting $t_1 = 3.0 \, \text{s}$ and $t_2 = 9.0 \, \text{s}$: $\frac{1}{2} a_x (3.0)^2 + a_x (3.0)(9.0) = 100$ $\frac{1}{2} a_x (9) + a_x (27) = 100$ $4.5 a_x + 27 a_x = 100$ $31.5 a_x = 100$ $a_x = \frac{100}{31.5}$ $a_x \approx 3.17 \, \text{m/s}^2$

So, the value of the acceleration $a_x$ is approximately $3.17 \, \text{m/s}^2$.

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Would you like more details or have any questions?

Here are some related questions you can explore:

1. What would be the final velocity of the sprinter after 3 seconds?
2. How much distance does the sprinter cover in the first 3 seconds?
3. What would happen if the acceleration phase lasted for a different time?
4. How much time would the sprinter need to finish the race if her acceleration was lower?
5. How does the distance-time graph look for this kind of motion?

Tip: Breaking motion into distinct phases helps simplify complex motion problems.