To determine the time required for a fast-moving vehicle to overtake slow-moving vehicles spaced 25 m apart, we need to calculate the distance covered during acceleration and the time taken to reach a certain speed.

Acceleration and Distance Calculation
- **Acceleration (a):** 1 m/s²
- **Initial Velocity (u):** 0 m/s (assuming the fast-moving vehicle starts from rest)
- **Final Velocity (v):** This is the speed needed to overtake the slow vehicles.
To calculate the distance (d) covered during acceleration until reaching a speed suitable for overtaking, we can use the formula:

d = ut + (1/2)at²
- Here, **u = 0**, so the formula simplifies to:

d = (1/2)at²

Overtaking Distance
- **Spacing between vehicles:** 25 m
- To safely overtake one vehicle, the fast-moving vehicle must cover the distance of at least 25 m.
Thus, setting d = 25 m:

25 = (1/2) * 1 * t²
This gives:

25 = 0.5t²
- Rearranging provides:

t² = 50

Time Calculation
- Solving for t:

t = √50 ≈ 7.07 seconds

Conclusion
The time required for the fast-moving vehicle to accelerate and safely overtake the slow-moving vehicles spaced 25 m apart is approximately **7.07 seconds**. This calculation assumes constant acceleration and ideal conditions without considering external factors like traffic or road conditions.