AERE_160_Midterm_Fall_23.pdf

AER E 160Midterm Fall 2023Due: October 14 th 11:59pm Please do the following problems. Show all your work. Most of the points for this assignment are for the problem-solving process. Work can be either handwritten or typed out, but all must be neatly done. Points deducted from difficult to read work will not be given back. Submit all your work as a single PDF on Canvas. The submission link on Canvas will only accept PDFs. You will not be able to upload non-PDF files in the submission link. This exam is open notes, book, and internet. Do not discuss your work with your classmates. You will not be able to get an extension on this assignment. Once the link closes on Canvas you will have no opportunity to submit the assignment for grading. Page 1 of 22

AER E 160Midterm Fall 2023Due: October 14 th 11:59pm Problem 1 (20 pts) A balloon in a university building at sea-level has a volume of . It is filled with perfectly safe 46 ?? 3 hydrogen. a.) In equilibrium, what is the mass of the payload? Use English units. (5 pts) Volume (V) = 46 ft^3 Density of Hydrogen ( ⍴ ) = 0.0051 lb/ft^3 Mass (m) = ? m = V * ⍴ m = 46 ft^3 * 0.0051 lb/ft^3 m = 0.23 lbs Therefore, the mass of the payload is equal to 0.23 lbs. b.) In equilibrium, what is the weight of the payload? Use English units. (5 pts) Acceleration due to gravity (g) = 32.2 ft/s^2 Weight (W) = ? W = m * g W = 0.23 lbs * 32.2 ft/s^2 W = 7.4 lbs Therefore, The weight of the payload is equal to 7.4 lbs. Page 2 of 22

AER E 160Midterm Fall 2023Due: October 14 th 11:59pm Problem 1 continued: c.) If the balloon was filled with helium, what is the weight of the payload in equilibrium? Use SI units. (5 pts) Volume = 46 ft^3 = 1.30 m^3 Density of Helium ( ⍴ ) = 0.18 kg/m^3 Acceleration due to gravity (g) = 9.8 m/s^2 Mass (m) = ? Weight (W) = ? m = ⍴ * V m = 0.18 kg/m^3 * 1.30 m^3 m = 0.23 kg W = m * g W = 0.23 kg * 9.8 m/s^2 W = 2.25 kg Therefore, the weight of the payload with Helium is equal to 2.25 kg Page 3 of 22

AER E 160Midterm Fall 2023Due: October 14 th 11:59pm d.) What volume of helium would be required to lift a 10 kg payload at sea-level? Use SI units (5 pt) Mass (m) = 10 kg Density ( ⍴ ) = 0.18 kg/m^3 Volume (V) = ? V = m / ⍴ V = 10 kg / 0.18 kg/m^3 V = 55.5 m^3 Therefore, the volume required to lift a 10 kg payload is equal to 55.5 m^3. Problem 2 (20 pts) – Standard Atmosphere a.) What is the difference between geometric, absolute, and geopotential altitudes? (6 pts) Geometric altitude, also referred to as geometric height, represents the actual straight-line distance above a reference point, often the Earth's surface or a reference ellipsoid. This measurement does not factor in changes in gravitational force or the Earth's shape. Absolute altitude, on the other hand, denotes the height above the Earth's surface measured along a line perpendicular to the reference ellipsoid. It does not consider variations in gravity's strength with altitude, making it less suitable for certain aerodynamic and atmospheric calculations. Geopotential altitude takes into account the gravitational potential energy per unit mass and acknowledges variations in gravitational force with altitude. It is the preferred choice for atmospheric calculations, providing a consistent measure of altitude even in regions with Page 4 of 22

AER E 160Midterm Fall 2023Due: October 14 th 11:59pm significant gravitational field variations. This measurement is typically expressed in units of length, such as meters or feet. b.) What is the difference between isothermal layers and gradient regions in Standard Atmosphere. (4pts) An isothermal layer in the atmosphere refers to a section where the temperature remains consistent as you go higher in altitude. In the Standard Atmosphere, there are areas where temperature doesn't alter as you ascend. These layers are primarily located in the lower atmosphere (troposphere) and the thermosphere. In gradient regions of the standard atmosphere, temperature changes with altitude, and this change is typically described using a temperature lapse rate. In these zones, temperature typically decreases as you move higher in the atmosphere, which is a common occurrence. The standard atmosphere model assumes a specific temperature lapse rate for these regions. c.) What is the value of the acceleration due to gravity (g) at 90,000ft? (5pts) Height (h) = 90,000 ft = 27.4 km Radius of Earth (R) = 6378.1 km Acceleration due to Gravity at Earth’s surface (g 0 ) = 9.8 m/s^2 Acceleration due to Gravity at a height (g(h)) = ? g(h) = g 0 (R / (R + h))^2 g(27.4) = 9.8 (6378.1 / (6378.1 + 27.4))^2 g (27.4) = 9.72 m/s^2 or 31.9 ft/s^2 Page 5 of 22

AER E 160Midterm Fall 2023Due: October 14 th 11:59pm Therefore, the acceleration due to gravity at 90,000 ft is equal to 31.9 ft/s^2 d.) Why do we use geopotential altitude instead of geometric altitude in Standard Atmosphere calculations? (5pts) Geopotential altitude is favored over geometric altitude because it offers a more reliable way to measure altitude in areas where gravity's strength varies more. While geometric altitude is simple, it doesn't consider the shifting gravitational force at different altitudes. Using geometric altitude in regions with significant gravity variations, like near the Earth's surface compared to higher altitudes, would cause inaccuracies. Therefore, geopotential altitude is the preferred measure for such calculations due to its consistency and accuracy in accounting for these variations in gravity. Page 6 of 22

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