Question

In a hydroelectric power station, the water is flowing at \(2 \, {ms}^{-1}\) in the river which is \(100 \, {m}\) wide and \(5 \, {m}\) deep. The maximum power output from the river is:

• A. \(1.5 \, {MW}\)
• B. \(2 \, {MW}\)
• C. \(2.5 \, {MW}\)
• D. \(3 \, {MW}\)

Hint:

The actual power output in hydroelectric plants depends significantly on the efficiency and height drop, which should be factored into realistic calculations.

Answer 1

The Correct Option is B

The maximum power output from the river can be calculated using the formula for the kinetic energy of flowing water. The power \( P \) is given by:

\[ P = \frac{1}{2} \rho A v^3 \]

where:

• \( \rho = 1000 \, \text{kg/m}^3 \) is the density of water,
• \( A \) is the cross-sectional area of the river,
• \( v = 2 \, \text{m/s} \) is the velocity of water.

The cross-sectional area \( A \) of the river is:

\[ A = \text{width} \times \text{depth} = 100 \times 5 = 500 \, \text{m}^2 \]

Now, substituting the values into the power formula:

\[ P = \frac{1}{2} \times 1000 \times 500 \times (2)^3 \] \[ P = \frac{1}{2} \times 1000 \times 500 \times 8 \] \[ P = 2000000 \, \text{W} = 2 \, \text{MW} \]

Thus, the maximum power output from the river is 2 MW.