A helicopter of mass 500kg hovers when its rotating blades move through an area of 45m². Find the average speed imparted to air.

To find the average speed imparted to the air by the helicopter's rotor blades, we can start by applying the principles of physics that relate to lift and momentum. Specifically, we will utilize the concepts of thrust and the area through which the rotor blades push the air. Let's break this down step by step.

Understanding the Forces at Play

When a helicopter hovers, it generates lift equal to its weight. The weight of the helicopter can be calculated using the equation:

Weight (W) = mass (m) × gravitational acceleration (g)

Here, the mass (m) is 500 kg, and the gravitational acceleration (g) is approximately 9.81 m/s².

Calculating the Weight

Now, substituting the values into the equation:

W = 500 kg × 9.81 m/s² = 4905 N

This means the helicopter must generate a thrust of 4905 N to hover.

Connecting Thrust to Airspeed

The thrust produced by the rotor blades is a result of the momentum change of the air. When the blades rotate, they push air downwards, creating an upward lift that counteracts gravity. The thrust can also be expressed in terms of the mass flow rate of the air and its velocity:

Thrust (T) = mass flow rate (ṁ) × velocity (v)

To find the average speed imparted to the air, we need to determine the mass flow rate of the air that the rotor blades interact with. This can be calculated using the area of the rotor blades and the density of air.

Estimating Mass Flow Rate

The mass flow rate can be expressed as:

ṁ = density (ρ) × area (A) × velocity (v)

For air, the density (ρ) is approximately 1.225 kg/m³ at sea level. The area (A) is given as 45 m². We can rearrange this equation to find the velocity:

v = Thrust / (ρ × A)

Finding the Average Speed

Now, let's substitute the known values into the rearranged equation:

v = 4905 N / (1.225 kg/m³ × 45 m²)

Calculating the denominator:

1.225 kg/m³ × 45 m² = 55.125 kg/s

Now, substitute this back into the equation for velocity:

v = 4905 N / 55.125 kg/s ≈ 89.0 m/s

Conclusion

The average speed imparted to the air by the helicopter's rotating blades is approximately 89.0 m/s. This speed reflects how fast the air is being pushed downwards by the rotor blades to generate sufficient lift for the helicopter to maintain its hover.