Is this project an undergraduate, graduate, or faculty project?
Undergraduate
Project Type
individual
Campus
Daytona Beach
Authors' Class Standing
Lukas Estrella, Freshman
Lead Presenter's Name
Lukas Estrella
Lead Presenter's College
DB College of Engineering
Faculty Mentor Name
Dr. Xuejian Li
Abstract
This project, "The Motion of a Falling Object Under Linear Drag and How Differential Equations Can Be Used To Find It," investigates the motion of a falling object subject to air resistance through a combination of mathematical modeling and fundamental physical principles. The analysis is grounded in Newton’s second law, which yields a differential equation describing the forces acting on the object. Assuming a linear drag model, in which the resistive force is proportional to velocity, the governing equation reduces to a first-order ordinary differential equation for velocity. This equation is solved using the integrating factor method, yielding an explicit expression for velocity as a function of time and a clear definition of terminal velocity as the steady-state solution. Building on this result, the velocity function is integrated to determine the object’s position over time, allowing for a complete description of its motion, while also examining how initial conditions influence both transient and long-term behavior. Although analytical solutions are obtained for the linear case, numerical methods are introduced to handle more complex scenarios, such as nonlinear drag or varying environmental conditions. Overall, this study demonstrates the power of differential equations in modeling real-world phenomena, particularly in capturing the effects of resistive forces, and highlights how idealized models can be refined to better approximate physical reality.
Did this research project receive funding support (Spark, SURF, Research Abroad, Student Internal Grants, Collaborative, Climbing, or Ignite Grants) from the Office of Undergraduate Research?
No
Included in
Dynamic Systems Commons, Fluid Dynamics Commons, Ordinary Differential Equations and Applied Dynamics Commons
The Motion of a Falling Object Under Linear Drag and How Differential Equations Can Be Used To Find It
This project, "The Motion of a Falling Object Under Linear Drag and How Differential Equations Can Be Used To Find It," investigates the motion of a falling object subject to air resistance through a combination of mathematical modeling and fundamental physical principles. The analysis is grounded in Newton’s second law, which yields a differential equation describing the forces acting on the object. Assuming a linear drag model, in which the resistive force is proportional to velocity, the governing equation reduces to a first-order ordinary differential equation for velocity. This equation is solved using the integrating factor method, yielding an explicit expression for velocity as a function of time and a clear definition of terminal velocity as the steady-state solution. Building on this result, the velocity function is integrated to determine the object’s position over time, allowing for a complete description of its motion, while also examining how initial conditions influence both transient and long-term behavior. Although analytical solutions are obtained for the linear case, numerical methods are introduced to handle more complex scenarios, such as nonlinear drag or varying environmental conditions. Overall, this study demonstrates the power of differential equations in modeling real-world phenomena, particularly in capturing the effects of resistive forces, and highlights how idealized models can be refined to better approximate physical reality.