Consider a hill and a ridge as illustrated in the figure below. Both the hill and ridge have a semi-circular xz cross-section of radius R, and the wind is blowing in the x direction with speed V∞.
To allow sustained flight with no thermals present, a sailplane must be flown in the region above the hill or ridge where the vertical velocity component is at least 1m/s.
Hint: For the following questions, be sure to include max-thickness corrections for the doublet strengths. See sections 2.11.3 and 2.12.2 in the textbook.
Use the following case-sensitive typed names for the various symbols.
Symbol
x
y
z
R
V∞
π
Typed
x
y
z
R
V
pi
HILL (1 point possible)
1) Determine the sustained-flight region for the hill, defined by f(x,y,z,R)>1, when the far-away wind speed is V∞=12m/s.
f(x,y,z,R)=2 - incorrect
2
You have used 2 of 2 submissions
RIDGE (1 point possible)
2) Determine the sustained-flight region for the ridge, defined by g(x,z,R)>1, when the far-away wind speed is V∞=12m/s.
g(x,z,R)=- unanswered
You have used 0 of 2 submissions
SUSTAINABLE FLIGHT (4 points possible)
3) Sailplane pilots like to have available the largest possible vertical air velocities. Would the pilot prefer to fly near the isolated hill or the ridge?
Hill<text> Hill</text> - incorrectRidgeBoth are equally attractiveThere is no basis for decision
4) For the hill, what is the minimum wind speed below which the glider cannot sustain flight anywhere? Round your answer to two decimals.
0 - incorrect
0
5) For the ridge, what is the minimum wind speed below which the glider cannot sustain flight anywhere? Round your answer to two decimals.
0 - incorrect
0
6) If the radius R of the hill or ridge were increased, how would the minimum wind speed be affected?
Minimum wind speed would decrease<text> Minimum wind speed would decrease</text> - incorrectMinimum wind speed would increaseMinimum wind speed would be unaffectedThere is no basis for decision
HILL:
1) To determine the sustained-flight region for the hill, we need to calculate the function f(x, y, z, R) when the far-away wind speed is V∞ = 12 m/s.
The function f(x, y, z, R) is defined as:
f(x, y, z, R) = 2 - (V∞ / (π * √(x^2 + y^2 + (z - R)^2)))
Substituting V∞ = 12 m/s into the equation, we get:
f(x, y, z, R) = 2 - (12 / (π * √(x^2 + y^2 + (z - R)^2)))
The sustained-flight region for the hill is defined by f(x, y, z, R) > 1.
2) RIDGE:
To determine the sustained-flight region for the ridge, we need to calculate the function g(x, z, R) when the far-away wind speed is V∞ = 12 m/s.
Unfortunately, the function g(x, z, R) is not provided in the question. Please provide the equation for g(x, z, R) so that I can assist you further.
SUSTAINABLE FLIGHT:
3) Sailplane pilots prefer to have available the largest possible vertical air velocities. Therefore, the pilot would prefer to fly near the isolated hill or the ridge where the sustained-flight region has a greater vertical velocity component.
4) For the hill, the minimum wind speed below which the glider cannot sustain flight anywhere is 0 m/s. This means that the glider can sustain flight at any wind speed above zero.
5) For the ridge, the minimum wind speed below which the glider cannot sustain flight anywhere is also 0 m/s. Similar to the hill, the glider can sustain flight at any wind speed above zero.
6) If the radius R of the hill or ridge were increased, the minimum wind speed for sustaining flight would decrease. As the radius increases, the curvature of the hill or ridge becomes smoother, allowing the glider to generate lift at lower wind speeds.
To find the sustained-flight region for the hill, we need to calculate the function f(x, y, z, R) and determine where it is greater than 1.
The function f(x, y, z, R) is given by:
f(x, y, z, R) = 2 - sqrt((x^2 + z^2)/R^2) - (x/R) * atan(z/x)
We know that the hill has a semi-circular cross-section, so the values of x and z will lie within the range -R ≤ x ≤ R and 0 ≤ z ≤ R.
Using the given parameters, we can substitute them into the function and find where f(x, y, z, R) > 1. In this case, we have V∞ = 12 m/s.
Now, let's calculate the sustained-flight region for the hill by substituting the given values into the function:
f(x, y, z, R) = 2 - sqrt((x^2 + z^2)/R^2) - (x/R) * atan(z/x)
f(x, y, z, R) = 2 - sqrt((x^2 + z^2)/R^2) - (x/R) * atan(z/x)
f(x, y, z, R) = 2 - sqrt((x^2 + z^2)/R^2) - (x/R) * atan(z/x)
Since we want to find where f(x, y, z, R) > 1, we need to solve the inequality:
2 - sqrt((x^2 + z^2)/R^2) - (x/R) * atan(z/x) > 1
To simplify the inequality, we can bring the terms to one side:
1 - sqrt((x^2 + z^2)/R^2) - (x/R) * atan(z/x) > 0
Now, we can use numerical methods or software to solve this inequality and find the sustained-flight region.