Understanding the Problem:

We are given that a sprinter has a maximum speed of 10 m/s and reaches that speed with an acceleration of 2.5 m/s². We need to determine the time it takes for the sprinter to run a distance of 100 m. To solve this problem, we will use the kinematic equation that relates distance, acceleration, final velocity, initial velocity, and time.

Solution:

Step 1: Identify the Known Values:
- Maximum speed (v) = 10 m/s
- Acceleration (a) = 2.5 m/s²
- Distance (d) = 100 m

Step 2: Identify the Kinematic Equation:
The kinematic equation that relates distance, acceleration, final velocity, initial velocity, and time is:
v² = u² + 2as

Where:
- v is the final velocity
- u is the initial velocity
- a is the acceleration
- s is the distance

Step 3: Rearrange the Equation:
Since we are given the final velocity, acceleration, and distance, we can rearrange the equation to solve for time (t). The rearranged equation becomes:
t = (v - u) / a

Step 4: Calculate the Time:
Substituting the given values into the equation, we have:
t = (10 m/s - 0 m/s) / 2.5 m/s²
t = 10 s / 2.5
t = 4 s

Step 5: Interpret the Result:
The sprinter takes 4 seconds to run a distance of 100 meters.

Summary:
- Given the sprinter's maximum speed of 10 m/s and an acceleration of 2.5 m/s², we calculated the time it takes for the sprinter to run 100 meters.
- Using the kinematic equation v² = u² + 2as, we rearranged the equation to solve for time as t = (v - u) / a.
- Substituting the known values, we found that the sprinter takes 4 seconds to run 100 meters.