1. Module 3.1 Modeling Falling and Skydiving
Angela B. Shiflet and George W. Shiflet
Wofford College
© 2014 by Princeton University Press

2. Model falling ball Simplification Variables and equations No friction
Position, s
Velocity, v(t) = ds/dt
Acceleration, a(t) = dv/dt
Acceleration due to gravity = m/sec2

3. Model falling ball

4. Refinement of model - friction
Forces acting on body
Weight
Newton’s Second Law, F = ma
F , force acting on body
m, mass of body
a, acceleration
unit for force - Newton (N) or kg m / sec2

5. Model falling ball Forces acting on body Kinetic friction or drag
Estimates
Stokes' friction, F = kv (for a very small object moving slowly through fluid)
k, kg/sec, constant of proportionality
v in m/sec, velocity.
Newtonian friction, F = 0.5CDAv2 (for a large object moving faster through a fluid)
C, constant of proportionality (coefficient of drag or drag coefficient); dimensionless constant related to the shape of the object
D, density of the fluid
A, object's projected area in the direction of movement

6. Model falling ball Forces acting on body
Kinetic friction or drag
Estimates
Stokes' friction, F = kv
Newtonian friction, F = 0.5CDAv2
Newtonian friction through air, F = 0.65Av2 (for large objects with C = 1 moving faster through sea-level air where the density of the air is D = 1.29kg/m3 at sea level)
A: object’s projected area in direction of movement
v: velocity
Note that F = 0.5CDAv2 can be rewritten as F = -0.5CDAv|v| (sine the drag force is in the opposite direction as v).

7. Model falling ball with air friction
Figure 3.1.4

8. Equation Set 3.1.1 DE: m*dv/dt = -m*g + 0.65*A*v2, v(0) = 0; this is
a non-linear DE.

9. Equation Set 3.1.1

10. Model skydiving
We ignore changes of air density for simplicity. Note the projected area A is not a constant here as in the falling ball example.

11. Model skydiving
The model for a skydive out of a helicopter has two phases, one where the person is in a free fall (A ~ 0.4 m2) and the other after the parachute opens, when the larger surface area results in more air resistance (A ~ 28 m2). Assume the pull of the ripcord is at 1000 m above the ground, and the parachute fully opens instantaneously.

12. Model skydiving