Module 3.1 Modeling Falling and Skydiving

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Presentation transcript:

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

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 = -9.81 m/sec2

Model falling ball

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

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

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).

Model falling ball with air friction Figure 3.1.4

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

Equation Set 3.1.1

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.

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.

Model skydiving