A train of mass 1600 kg attains a speed of 25 ms$^{-1}$ in 20 seconds. The power developed in the engine is?

Analysis

To find the power developed by the engine, we need to understand the relationship between power, force, and velocity. Power is defined as the rate of doing work, or the product of force and velocity ($P = F \times v$).

Concept

1. Force ($F$): According to Newton’s Second Law, $F = ma$. Since the train starts from rest ($u = 0$) and reaches a final velocity ($v = 25 \text{ ms}^{-1}$) in time ($t = 20 \text{ s}$), we first find the acceleration ($a$): $$a = \frac{v - u}{t}$$
2. Power ($P$): Since the velocity is changing, we use the average velocity ($v_{avg}$) to calculate the power developed over that interval: $$P = F \times v_{avg}$$Where$v_{avg} = \frac{u + v}{2}$

Step-by-Step Solution

Step 1: Calculate the acceleration ($a$)

Step 2: Calculate the Force ($F$)

Step 3: Calculate the average velocity ($v_{avg}$)

Step 4: Calculate the Power ($P$)

Step 5: Convert to Kilowatts (KW) Since $1000 \text{ W} = 1 \text{ KW}$:

Conclusion

The power developed by the engine is 25 KW. This matches option C.

Exam Tip

When a question involves acceleration from rest, remember that the power developed is not constant. Always use the average velocity ($v_{avg} = \frac{v+u}{2}$) when calculating the power developed during a period of uniform acceleration. If you had used the final velocity ($25 \text{ ms}^{-1}$) instead of the average, you would have arrived at 50 KW, which is a common distractor in JAMB questions!