Operation Desert Stormhawk

OPERATION DESERT STORMHAWK

Master the critical calculations needed to ensure a life-saving payload reaches its target location.

The Problem

Identify The Problem

A severe disease outbreak has struck the village of Watapur in Kunar Province, Afghanistan, and immediate medical intervention is critical to saving lives. With a population of 2,500 people at an elevation of 1,000 meters, Watapur is difficult to reach by ground, and the surrounding roads are obstructed due to ongoing conflicts. The nearest hospital is in Asadabad, 10 kilometers away, but the roads are controlled by hostile forces, leaving the village isolated and in urgent need of help.

The only hope for delivering the life-saving medical supplies—antibiotics, IV fluids, and antiviral medications—is through an aerial drop by the MQ-9 Reaper drone. Captain Mason has tasked Lieutenant Blake with ensuring the drone can successfully deliver these supplies to an open field near the Pech River in Watapur, avoiding the rough terrain that could damage or lose the cargo.

In this mission, you will step into Lieutenant Blake’s shoes and perform the necessary calculations to ensure the Reaper drone’s success. You’ll need to:

Calculate Takeoff Requirements Using the Third Kinematic Equation:
To ensure the drone can safely take off from both the USS Dwight D. Eisenhower and Bagram Air Base, you will use the third kinematic equation, which relates final velocity, initial velocity, constant acceleration, and displacement. This will allow you to verify whether the drone can achieve the necessary speed for takeoff given the constraints of each location.
Calculate the Coefficient of Lift:
Given the drone’s speed and other variables such as air density and wing area, students will use the coefficient of lift formula. This calculation will help determine if the drone generates enough lift for safe takeoff.
Convert Rates and Units:
Use proper unit analysis to convert rates and measurements, ensuring that all variables, such as fuel consumption and payload weight, are expressed in consistent units for accurate calculations.
Solve Cubic Equations by Graphing Using Technology Tools:
To find the angle of attack for safe flight, students will solve cubic equations by graphing. Using a graphing utility such as Desmos, students will model the relationship between the angle of attack and lift coefficient. This technology will help them ensure that the angle of attack remains within safe limits.
Solve a Quadratic Equation Using the Quadratic Formula:
In the process of finding the stall angle, students will use the quadratic formula to solve a quadratic equation related to the lift coefficient. This quadratic equation is the result of optional work with derivatives, but calculus is not necessary to complete the task.
Determine the Precise Payload Drop Coordinates Using Google Earth:
As part of determining the payload drop coordinates, students will use Google Earth to locate the precise latitude and longitude over the Pech River where the payload should be released. In addition, students will solve a quadratic equation using square roots to calculate the horizontal distance the payload will travel. This step ensures that the supplies land in the correct location, taking into account only the drone’s flight path and the terrain.
Analyze Variables:
Account for changes in air density, altitude, and payload weight to guarantee a successful mission from both takeoff locations.

By the end of this lesson, you’ll have a better understanding of how to use mathematical models and technology tools to analyze real-world scenarios and draw meaningful conclusions for complex problem-solving.

Data, Variables, And Assumptions

In this lesson, we focus on a critical mission involving an MQ-9 Reaper drone tasked with delivering life-saving medical supplies to the remote village of Watapur in Afghanistan. The calculations and analyses we perform will help ensure the drone’s safe takeoff, flight, and accurate delivery of the payload. Below are the key data, variables, and assumptions we’ll be working with in this scenario.

Data Accuracy

We have made every effort to ensure that the data used in this lesson is as accurate as possible, including detailed specifications of the MQ-9 Reaper drone. The geographic data, including locations and elevations, is based on reliable sources and reflects real-world terrain in Afghanistan.

However, certain aspects of the data, such as the takeoff speed and acceleration of the drone, are estimated for this lesson. Additionally, while we do not have access to the exact polynomial that models the relationship between the coefficient of lift (CL) and the angle of attack (alpha) for the Reaper, we have attempted to model one based on real-world data. This allows us to simulate the flight conditions while maintaining a focus on the core mathematical principles.

Data and Sources

The following data will be relevant for our calculations:

Geographical Data:
- Watapur, Afghanistan: Population of 2,500 people, elevation of 1,000 meters.
- Bagram Air Base: Located approximately 100 kilometers away from Watapur.
- Pech River: The release point for the drone’s payload will be over this river near Watapur.
Drone Specifications:
- MQ-9 Reaper:
  - Maximum speed: 300 mph.
  - Fuel capacity: 600 gallons.
  - Wing area: 17.1 square meters.
  - Coefficient of lift (CL) vs angle of attack (alpha) polynomial: An estimated model based on real-world data.
  - Source: United States Air Force, “MQ-9 Reaper,” U.S. Air Force Fact Sheet, https://www.af.mil/About-Us/Fact-Sheets/Display/Article/104470/.
Flight Conditions:
- Takeoff velocity at Bagram Air Base: Estimated at 51.4 meters per second.
- Acceleration: Estimated at 4.4 m/s². While this acceleration may be too high for actual takeoff conditions, it is used for the purposes of this lesson to simplify the calculations.

Variables

The key variables we will work with include:

Velocity:
The speed of the drone during takeoff and during the flight to Watapur. This will impact the calculations for lift and angle of attack.
Acceleration:
The drone’s acceleration during takeoff, estimated at 4.4 m/s² for this lesson. This will be used in calculating the drone’s final velocity using the third kinematic equation. Although this value may be higher than typical, it is employed to simplify the lesson’s calculations.
Angle of Attack:
The angle at which the drone’s wings meet the oncoming air. This will be analyzed by solving cubic equations to ensure it stays below the stall angle.
Coefficient of Lift:
Using the estimated lift coefficient model, students will calculate the lift generated by the drone’s wings based on its velocity and wing area.
Displacement (Runway Length):
The estimated distance the drone travels along the runway before takeoff is 300 meters. This displacement will be used in conjunction with the drone’s acceleration and velocity to ensure it achieves the necessary takeoff speed.
Fuel Load:
The weight of the fuel onboard, which affects the total takeoff weight of the drone and is accounted for in lift and takeoff calculations.
Payload Drop Coordinates:
The exact latitude and longitude of the release point over the Pech River, calculated using Google Earth and trajectory equations to ensure the payload lands safely in the designated open area.

Assumptions

Air Density at Bagram:
The air density at Bagram Air Base is estimated to be 1.15 kg/m³ for the purposes of this lesson. This is a reasonable estimate based on the elevation and environmental conditions.
No Wind Factor:
For this lesson, wind resistance and effects on the payload’s trajectory are not taken into consideration, focusing instead on the mathematical relationships between velocity, altitude, and displacement.
Simulated Data:
The flight performance of the MQ-9 Reaper and the geographical information used in this lesson are based on real-world data, but the scenario is a simulation to allow students to apply mathematical models without relying on classified or sensitive information.

Why Do We Use These Assumptions?

By using estimated and simulated data, students can focus on learning and applying mathematical principles in a realistic scenario without needing access to sensitive or classified information. The estimated takeoff speeds and acceleration provide a reasonable approximation, and the modeled CL vs alpha relationship allows for meaningful analysis of flight dynamics while practicing polynomial calculations and graphing.