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

2. 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

3. 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

4. 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)

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

6. Solution Pictures

8. Maximum Height When vertical velocity vanishes
Position at that time is

9. Range

10. 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)

11. Vertical Case
Integrate to get the position

12. Taylor Picture (Sign Reversed)

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

14. Results