A general aviation aircraft (m = 1200 kg) flies (under ISA conditions) at 850 metres altitude, with a constant velocity (true airspeed) of 116.6 knots. Its wing surface area is 22 square metres.
Given that its propeller is able to accelerate 80 kilograms of air to a velocity of 105.56 m/s every second, determine the plane's drag coefficient .
To determine the plane's drag coefficient, we need to calculate the drag force acting on the aircraft. The drag force can be calculated using the following equation:
Drag force = 0.5 * air density * velocity^2 * wing surface area * drag coefficient
First, let's calculate the air density at 850 meters altitude. The air density at a specific altitude can be found using the International Standard Atmosphere (ISA) model, which states that the air density decreases at a rate of approximately 1.225 kg/m^3 per 1000 meters.
Air density at 850 meters altitude = (1.225 kg/m^3 - 0.001225 kg/m^3 * 0.85) = 1.043 kg/m^3
Next, let's convert the true airspeed from knots to meters per second:
True airspeed in meters per second = 116.6 knots * 0.5144 m/s = 59.98 m/s
Now, we can calculate the drag force:
Drag force = 0.5 * 1.043 kg/m^3 * (59.98 m/s)^2 * 22 m^2 * drag coefficient
To find the drag coefficient, we need to rearrange the equation:
drag coefficient = drag force / (0.5 * 1.043 kg/m^3 * (59.98 m/s)^2 * 22 m^2)
Given that the propeller accelerates 80 kilograms of air to a velocity of 105.56 m/s every second, the force exerted by the propeller is equal to the change in momentum per second:
Force = mass * velocity change
Force = 80 kg * (105.56 m/s - 0 m/s) / 1 second = 8444.8 kg m/s^2 = 8444.8 N
Since the propeller force is equal to the drag force, we have:
8444.8 N = drag force
Now, substituting this into the drag coefficient equation:
drag coefficient = 8444.8 N / (0.5 * 1.043 kg/m^3 * (59.98 m/s)^2 * 22 m^2)
In order to determine the plane's drag coefficient, we need to calculate the drag force. The drag force can be calculated using the following formula:
Drag force = (1/2) * ρ * v^2 * Cd * A
Where:
ρ = air density
v = velocity of the aircraft
Cd = drag coefficient
A = wing surface area
First, let's convert the velocity from knots to meters per second:
116.6 knots = 60.27 m/s
Given that the plane's altitude is 850 meters, we can use the International Standard Atmosphere (ISA) to find the air density at that altitude. The air density at sea level (ρ0) is approximately 1.225 kg/m^3, and it decreases with altitude at a rate of -0.0065 kg/m^3 per meter. We can use this information to calculate the air density at 850 meters:
ρ = ρ0 * (1 + (L * h / T0)) ^ (-g / (L * R))
Where:
L = temperature lapse rate (0.0065 K/m)
h = altitude (850 m)
T0 = sea level temperature (288.15 K)
g = acceleration due to gravity (9.81 m/s^2)
R = specific gas constant for air (287.1 J/kg·K)
Using this equation, we find:
ρ = 1.225 * (1 + (0.0065 * 850 / 288.15)) ^ (-9.81 / (0.0065 * 287.1))
= 1.082 kg/m^3
Now we can calculate the drag force. Rearranging the formula for drag force, we can solve for Cd:
Drag force = (1/2) * ρ * v^2 * Cd * A
Cd = Drag force / ((1/2) * ρ * v^2 * A)
The mass of air accelerated by the propeller every second is 80 kg, and its velocity is 105.56 m/s. This mass and velocity represent the force produced by the propeller (thrust). At constant velocity, this thrust force must be equal to the drag force. Therefore, we can set the thrust force equal to the drag force:
Thrust force = Drag force
80 kg * acceleration = (1/2) * ρ * v^2 * Cd * A
Rearranging this equation, we can solve for the drag coefficient:
Cd = (2 * 80 kg * acceleration) / (ρ * v^2 * A)
Plugging in the given values:
Cd = (2 * 80 kg * 105.56 m/s) / (1.082 kg/m^3 * (60.27 m/s)^2 * 22 m^2)
Calculating this expression gives us the value for the drag coefficient Cd.
To determine the drag coefficient of the plane, we need to calculate the drag force acting on the aircraft. We can use the following equation:
Drag Force = 0.5 * Density * Velocity^2 * Drag Coefficient * Wing Surface Area
First, let's calculate the air density at 850 meters altitude. The International Standard Atmosphere (ISA) model can be used to approximate the air density at different altitudes. At sea level (0 meters altitude), the air density is approximately 1.225 kg/m^3. The density decreases by about 1.98% per 1000 meters increase in altitude under ISA conditions.
Density at 850 meters altitude:
Density = (1 - (0.0198 * (850 / 1000))) * 1.225
Density = 1.015 kg/m^3
Next, let's convert the true airspeed from knots to meters per second.
True Airspeed = 116.6 knots * 0.5144 m/s per knots
True Airspeed = 59.98 m/s
Now, let's calculate the drag force:
Drag Force = 0.5 * 1.015 kg/m^3 * (59.98 m/s)^2 * Drag Coefficient * 22 m^2
To solve for the drag coefficient, we rearrange the equation and isolate the Drag Coefficient:
Drag Coefficient = Drag Force / (0.5 * 1.015 kg/m^3 * (59.98 m/s)^2 * 22 m^2)
Now, let's calculate the drag coefficient:
Drag Coefficient = ((80 kg * 105.56 m/s) / (0.5 * 1.015 kg/m^3 * (59.98 m/s)^2 * 22 m^2))
Simplifying the equation further:
Drag Coefficient = (8444.8 N) / (0.5 * 1.015 kg/m^3 * 2154.1456 m^2)
Drag Coefficient = 8444.8 N / 1095.25344 kg*m/s^2 = 7.715
Therefore, the drag coefficient of the plane is approximately 7.715.