Question

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): Wind power generation depends on cube of wind speed.
Reason (R): Kinetic Energy of wind is proportional to the square of wind speed.
In the light of the above statements, choose the most appropriate answer from the options given below:

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

For wind energy numerical problems, always remember the important relation: \[ P = \frac{1}{2}\rho A v^3 \] A small increase in wind speed causes a very large increase in generated power because of the cube relationship.

Answer 1

The Correct Option is B

Concept: Wind energy systems convert the kinetic energy available in moving air into mechanical energy and then into electrical energy. The amount of power that can be extracted from the wind depends upon the mass of air flowing per unit time and the kinetic energy possessed by that air. The kinetic energy of a moving body is given by: \[ KE = \frac{1}{2}mv^2 \] where:
• \(m\) = mass of the body
• \(v\) = velocity of the body For wind turbines, the power available in wind is: \[ P = \frac{1}{2}\rho A v^3 \] where:
• \(\rho\) = density of air
• \(A\) = swept area of turbine blades
• \(v\) = wind velocity Thus, wind power depends on the cube of wind velocity.

Step 1: Understanding Assertion (A)
The assertion states that: \[ \text{Wind power generation depends on cube of wind speed} \] The mathematical expression for wind power is: \[ P = \frac{1}{2}\rho A v^3 \] From this expression, it is very clear that wind power is directly proportional to the cube of wind speed. That means: \[ P \propto v^3 \] This implies:
• If wind speed doubles, power becomes \(2^3 = 8\) times.
• If wind speed triples, power becomes \(3^3 = 27\) times. Hence, Assertion (A) is absolutely correct.

Step 2: Understanding Reason (R)
The reason states: \[ \text{Kinetic Energy of wind is proportional to square of wind speed} \] We know that kinetic energy is given by: \[ KE = \frac{1}{2}mv^2 \] Thus: \[ KE \propto v^2 \] Therefore, the reason statement is also correct.

Step 3: Checking Whether Reason Explains Assertion
Although both statements are individually correct, the reason does not completely explain why wind power depends on cube of velocity. The cube relationship arises because:
• Kinetic energy contributes one factor of \(v^2\)
• Mass flow rate of air also depends on velocity \(v\) Mass flow rate is: \[ \dot{m} \propto v \] Therefore: \[ P = \text{Mass flow rate} \times \text{Kinetic Energy} \] \[ P \propto v \times v^2 \] \[ P \propto v^3 \] Hence, the reason only explains the kinetic energy dependence and not the additional velocity factor due to mass flow rate. Therefore, Reason (R) is not the correct explanation of Assertion (A). Final Conclusion:
• Assertion (A) is correct.
• Reason (R) is also correct.
• But Reason (R) does not completely explain Assertion (A). Therefore, the correct answer is: \[ \boxed{\text{(B) Both (A) and (R) are correct but (R) is not the correct explanation of (A)}} \]