Physics 319 Classical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 3 G. A. Krafft Jefferson Lab

Drag Qualitatively Drag is the force opposing motion through a resisting medium. Example: air resists the motion of objects travelling through it. The drag force vanishes when the velocity vanishes is directed against the velocity can have different functional dependence on the velocity depending on the size and shape of the object and the properties of the resisting medium We will take cases dealing with air resistance, but the general methods apply to a wider variety of situations In this course will not consider cases, like an airplane’s wing, where the force (lift) is in a direction other than opposing the velocity

Quantitative Model Drag force First example of a velocity dependent force Taylor expanding b and c are called the linear and quadratic drag coefficient depend on size of the object for small objects generally linear term dominates for large objects the quadratic term is more important dividing line when the Reynold’s number < 1

Linear Drag Solvable Analytically Equations of motion The equations of motion are separately solvable in Cartesian coordinates Terminal velocity (velocity after wait much longer than m/b)

Initial conditions Using the solutions at t = 0 Trajectory starting at the origin

Solution Pictures

Maximum Height When vertical velocity vanishes Position at that time is

Range

Quadratic Drag Can solve analytically only the separate 1-D cases. First do horizontal case Separation of variables calculation Applying initial conditions Doesn’t damp so quickly (inverse power instead of exponential)

Vertical Case Integrate to get the position

Taylor Picture (Sign Reversed)

Two Dimensional Calculation Equations of motion now Or the now coupled equations Taylor used a computer

Results