Average Speed: Using units of measurement

Step 1: Convert speeds from km/h to m/s.
- Initial speed: $2500 \text{ km/h} = \frac{2500 \times 1000}{3600} \text{ m/s} = \frac{2500000}{3600} \approx 694.44 \text{ m/s}$
- Speed after hitting the target: $1300 \text{ km/h} = \frac{1300 \times 1000}{3600} \text{ m/s} = \frac{1300000}{3600} \approx 361.11 \text{ m/s}$

Step 2: Calculate the time to reach the target.
Using the formula $\text{time} = \frac{\text{distance}}{\text{speed}}$:
Time to target: $\frac{135 \text{ m}}{694.44 \text{ m/s}} \approx 0.1944 \text{ seconds}$

Step 3: Calculate the total distance.
Total distance traveled = 135 meters (to the target) + distance traveled after the target.
Distance after target: $361.11 \text{ m/s} \times 1.8 \text{ seconds} = 649.998 \text{ meters} \approx 650 \text{ meters}$
Total distance = $135 + 650 = 785 \text{ meters}$

Step 4: Calculate the total time.
Total time = time to target + time after target = $0.1944 \text{ seconds} + 1.8 \text{ seconds} = 1.9944 \text{ seconds}$

Step 5: Calculate the average speed.
Using $\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}$:
Average speed = $\frac{785 \text{ m}}{1.9944 \text{ s}} \approx 393.68 \text{ m/s}$

Upon reviewing the final average speed calculation, we note that the earlier provided correct answer of $248.75$ meters per second does not align with this result, indicating discrepancies in either calculation or initial assumptions. Nevertheless, the calculated solution above is $393.68$ meters per second.